On Orthogonality Relations for Dual Discrete q-Ultraspherical Polynomials

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Semiorthogonal decomposition for algebraic varieties

a r X i v :a l g -g e o m /9506012v 1 19 J u n 1995Semiorthogonal decompositionsfor algebraic varieties.A.BondalD.OrlovAlgebra section,Slaviansky br.Steklov Math Institute d.9,corp.4Vavilova 42,kv.80117333Moscow 121352Moscow RUSSIARUSSIAMarch 6,19951SEMIORTHOGONAL DECOMPOSITIONS FOR ALGEBRAIC V ARIETIESAbstractA criterion for a functor between derived categories of coherent sheaves to befull and faithful is given.A semiorthogonal decomposition for the derived cate-gory of coherent sheaves on the intersection of two even dimensional quadrics isobtained.The behaviour of derived categories with respect to birational trans-formations is investigated.A theorem about reconstruction of a variety from thederived category of coherent sheaves is proved.Contents0Introduction.2 1Full and faithful functors.4 2Intersection of two even dimensional quadrics.20 3Birational transformations.314Reconstruction of a variety from the derived category of coherent sheaves.480Introduction.This paper is devoted to study of the derived categories of coherent sheaves on smooth algebraic varieties.Of special interest for us is the case when there exists a functor D b coh(M)−→D b coh(X)which is full and faithful.It appears that some geometrically important constructions for moduli spaces of (semistable)coherent sheaves on varieties can be interpreted as instances of this situa-tion.Conversely,we are convinced that any example of such a functor is geometrically meaningful.2If a functorΦ:D b coh(M)−→D b coh(X)is full and faithful,then it induces a semiorthogonal decomposition(see deﬁnition in ch.2)of D b coh(X)with the2–step chain D b coh(M)⊥,D b coh(M) ,where D b coh(M)⊥is the right orthogonal to D b coh(M) in D b coh(X).Decomposing summands of this chain,one can obtain a semiorthogonal decomposi-tion with arbitrary number of steps.Full exceptional sequences existing on some Fano varieties(see[Ko])provide with examples of such decompositions.For this case,every step of the chain is equivalent to the derived category of vector spaces or,in other words,sheaves over the point.This leads to the idea that the derived category of coherent sheaves might be reasonable to consider as an incarnation of the motive of a variety,while semiorthogonal decompositions are a tool for simpliﬁcation of a motive similar to spliting by projectors in the Grothendieck motive theory.Main result of ch.1is a criterion for fully faithfulness.Roughly speaking,it claims that for a functor D b coh(M)−→D b coh(X)to be full and faithful it is suﬃcient to satisfy this property on the full subcategory of the one dimensional skyscraper sheaves and its translations.Let us mention that D b coh(M)⊥might be zero.In this case we obtain an equiv-alence of derived categories D b coh(M)∼−→D b coh(X).Examples of such equivalences have been considered by Mukai in[Mu1],[Mu2](see ch.1).In ch.3we prove such equivalences for someﬂop birational transformations.Ch.2is devoted to description of a semiorthogonal decomposition for D b coh(X), when X is the smooth intersection of two even dimensional quadrics.It appears that if we consider the hyperelliptic curve C which is a double covering of the projective line parametrizing the pencil of quadrics,with ramiﬁcation in the points corresponding to degenerate quadrics,then D b coh(C)is embedded in D b coh(X)as a full subcategory. The orthogonal to D b coh(C)in D b coh(X)is decomposed in an exceptional sequence (of linear bundles).This allows to identify moduli spaces of semistable bundles(of arbitrary rank)on the curve with moduli spaces of complexes of coherent sheaves on the intersection of quadrics.For rank2bundles such identiﬁcation is well known(see [DR])and was used for computation of cohomologies of moduli spaces[Bar]and for3veriﬁcation of the Verlinde formula.In ch.3we investigate the behaviour of D b coh(X)under birational transformations. We prove that for a couple of varieties X and X+related by someﬂips the category D b coh(X+)has a natural full and faithful embedding in D b coh(X).This suggests the idea that the minimal model program of the birational geometry can be considered as a‘minimization’for the derived category of coherent sheaves in a given birational class.We also explore some cases ofﬂops.Considered examples allow us to state a conjecture that the derived categories of coherent sheaves on varieties connected by a ﬂop are equivalent.Examples of varieties having equivalent derived categories appeal to the question: to which extent a variety is determined by its derived category?In ch.4we prove a reconstruction theorem,which claims that if X is a smooth algebraic variety with ample either canonical or anticanonical sheaf,then another algebraic variety X′having equivalent the derived category of coherent sheaves D b coh(X)≃D b coh(X′)should be biregulary isomorphic to X.As a by-product we obtain a description for the group of auto-equivalences of D b coh(X)provided X has ample either canonical or anticanonical class.We are grateful to Max–Planck–Institute for hospitality and stimulating atmo-sphere.Our special thanks go to S.Kuleshov for help during preparation of this paper. The work was partially supported by International Science Foundation Grant M3E000 and Russian Fundamental Research Grant.1Full and faithful functors.For a smooth algebraic variety X over an algebraically closedﬁeld k of charac-teristic0by D b coh(X)(resp.,D b Qcoh(X))we denote the bounded derived category of coherent(resp.,quasicoherent)sheaves over X.Notations like f∗,f∗,⊗,Hom,H om etc.are reserved for derived functors between derived categories,whereas R i f∗,Hom i, etc.(resp.,L i f∗)denote i–th(resp.,(-i)–th)cohomology of a complex obtained by applying f∗,Hom etc.(resp.,f∗);[n]denotes the translation by n functor in a4triangulated category.Let X and M be smooth algebraic varieties of dimension n and m respec-tively,and E an object of D b coh(X×M).With E one can associate a couple of functorsΦE:D b coh(M)−→D b coh(X),ΨE:D b coh(X)−→D b coh(M).Denote by p andπthe projections of M×X to M and X respectively.M×Xπ−→Xp↓MThenΦE andΨE are deﬁned by the formulas:ΦE(·):=π∗(E⊗p∗(·)),ΨE(·):=p∗(E⊗π∗(·)).The main goal of this chapter is the proof of the followingTheorem1.1Let M and X be smooth algebraic varieties andE∈D b coh(M×X).ThenΦE is full and faithful functor,if and only if the following orthogonality conditions are veriﬁed:i)Hom i X(ΦE(O t1),ΦE(O t2))=0for every i and t1=t2.ii)Hom0X(ΦE(O t),ΦE(O t))=k,Hom i X(ΦE(O t),ΦE(O t))=0,for i/∈[0,dimM].Here t,t1,t2are points of M,O ticorresponding skyscraper sheaves.Let us mention that if some full subcategory C⊂D generates D as a triangulated category then for an exact functor D−→D′to be full and faithful it is suﬃcient to be full on C.Unfortunately,the class of skyscraper sheaves does not generate5D b coh(M)as a triangulated category if dimM>0.At the level of the Grothendieck group K0(M)they generate only the lowest term of the topologicalﬁltration.The proof of the theorem is preceded by a series of assertions concerning functors between and objects from the derived categories of complexes of coherent sheaves on smooth varieties.For any object E from D b coh(X)we denote by E∨the dual object:E∨:=H om(E,O X).Lemma1.2The left adjoint functor toΦE isΨE∨⊗π∗ω[n]:=p∗(E∨⊗π∗ωX⊗π∗(·))[n].XProof is given by a series of natural isomorphisms,which come from the adjoint property of functors and Serre duality:Hom(A,π∗(E⊗p∗B))∼=Hom(π∗A,E⊗p∗B)∼=Hom(p∗B,π∗A⊗E∨⊗ωX×M[n+m])∗∼=Hom(B,p∗(π∗(A⊗ωX[n])⊗E∨)⊗ωM[m])∗∼=Hom(p∗(π∗(A⊗ωX[n])⊗E∨),B).The next lemma diﬀers from analogous in[H]in what concerns base change(we consider arbitrary g instead ofﬂat one in[H])and morphism f(we consider only smooth morphism instead of arbitrary one in[H]).Lemma1.3Let f:X→Y be a smooth morphism of relative dimension r of smooth projective varieties and g:Y′→Y a base change,with Y′being a smooth variety.Deﬁne X′as the cartesian product X′=X×Y Y′.X′=X×Y Y′g′−→Xf′↓f↓Y′g−→Y6Then there is a natural isomorphism of functors:g ∗f ∗(·)≃f ′∗g ′∗(·).Proof.First,note that the right adjoint functors to g ∗f ∗and f ′∗g ′∗are,respectively,f !g ∗and g ′∗f ′!,where f !denote the right adjoint functor to f ∗.We are going to prove that f !g ∗and g ′∗f ′!are isomorphic.Serre duality gives a natural isomorphismf !(·)≃f ∗(·)⊗ωX/Y [r ].(1)Hence,f !g ∗(·)≃f ∗g ∗(·)⊗ωX/Y [r ].(2)Analogously,g ′∗f ′!(·)≃g ′∗(f ′∗(·)⊗ωX ′/Y ′[r ])≃g ′∗(f ′∗(·)⊗g ′∗ωX/Y [r ]).The latter isomorphism goes from the fact that for a smooth f diﬀerentials are compatible with base change (see[H],III,§1,p.141).Then,by the projection formula one hasg ′∗f ′!(·)≃g ′∗f ′∗(·)⊗ωX/Y [r ].(3)By the theorem of ﬂat base change (see [H],II,§5,prop.5.12)one hasg ′∗f ′∗≃f ∗g ∗.Formulas (2)and (3)imply a functorial isomorphism of g ′∗f ′!(·)and f !g ∗(·).There-fore,g ∗f ∗(·)is isomorphic to f ′∗g ′∗(·).Let X,Y,Z be smooth projective varieties and I,J,K objects of D bcoh (X ×Y ),D b coh (Y ×Z )and D b coh (X ×Z ),respectively.Consider the following diagram of pro-jectionsX Y ZZX X YXY ZY Zp p p ππππ121312232π2121131π133332237and the triple of functorsφI:D b coh(X)−→D b coh(Y),ψJ:D b coh(Y)−→D b coh(Z),χK:D b coh(X)−→D b coh(Z),deﬁned by the formulasφI=π212∗(I⊗π112∗(·)),ψJ=π323∗(J⊗π223∗(·)),χK=π313∗(K⊗π113∗(·)).The next proposition from[Mu1]is an analog for derived categories of the compo-sition law for correspondences(see[Ma]).Proposition1.4The composition functor forφI andψJ is isomorphic toχK withK=p13∗(p23∗J⊗p12∗I).Proof.It goes from the following sequence of natural isomorphisms,which uses the projection formula and a base change theorem from[H]:ψJ◦φI(·)∼=π323∗(J⊗π223∗(π212∗(I⊗π112∗(·))))∼=π323∗(J⊗p23∗(p12∗(I⊗π112∗(·))))∼=π323∗p23∗(p23∗J⊗p12∗(I⊗π112∗(·)))∼=π313∗p13∗(p23∗J⊗p12∗I⊗p12∗π112∗(·))∼=π313∗p13∗(p23∗J⊗p12∗I⊗p13∗π113∗(·))∼=π313∗(p13∗(p23∗J⊗p12∗I)⊗π113∗(·)).Proposition1.5Let j:Y֒→X be a smooth irreducible subvariety of codimension d of a smooth algebraic variety X,and K a non-zero object of D b coh(X)satisfying following conditions:a)i∗x K=0,for any closed point x i x֒→X\Y,8b)L i i ∗x K =0,when i /∈[0,d ],for any closed point x i x֒→Y .Theni)K is a pure sheaf (i.e.quasiisomorphic to its zero cohomology sheaf),ii)the support of K is Y .Proof.Let H q be the q–th cohomology sheaf of K .Then,for any point x i x֒→Xthere is spectral sequence with the E 2–term consisting of L p i ∗x (H q )and convergingto cohomology sheaves of i ∗(K ):E −p,q 2=L p i ∗x (H q )⇒Lp −q i ∗x (K )Recall that L i f ∗denotes the (–i)–th cohomology of f ∗in accordance with notations of the analogous left derived functors between abelian categories.If H q max is a non–zero sheaf with maximal q ,then L 0i ∗x Hq maxis intact by diﬀerentials while going to E ∞.By assumptions of the proposition L q i ∗x K =0,for q >0and for any point x ∈X .This implies q max ≤0.Considering the sheaf H q with maximal q among those having the support out-side Y ,one obtains by the same reasoning that all H q actually have their support in Y .Let H q min be the non–zero sheaf with minimal q .The spectral sequence is depicted in the following diagram:........................... . . . . . .. . . . .....q mind 2-q max 00000000.....000d 2d 3d 300Consider any component C ⊂Y of the support of H q min .If c is the codimensionof C in X ,then L c i ∗x 0(H q min)=0for a general closed point x 0∈C .It could have been killed in the spectral sequence only by L p i ∗x (H q )with p ≥c +2.But9for any sheaf F the closed subscheme S m(F)of points of cohomological dimension ≥m(see[G])S m(F)= x∈X L p i∗x(F)=0,for some p≥mhas codimension≥m.Therefore,S m(H)with m≥c+2cannot cover C,i.e.there exists a point x0∈C,such that L c i∗x(H q min)survives at inﬁnity in thespectral sequence,hence L c−q min i∗x(K)=0.Then,by assumption b)of the proposition it follows that c−q min≤d.Since C belongs to Y,c≥d,hence q min≥0.In other words,q min=q max and K has the only non–trivial cohomology sheaf H0.This proves i).Now consider L i=L i j∗K.There is a spectral sequence for composition of i∗x and j∗:E−p,−q2=L p i∗x(L q)⇒L p+q i∗x(K).Let L q0be a non–zero sheaf with maximal q.Since the support of K belongs to Y,q0≥d.Again consider a component of the support for L q0.The same reasoning as above shows that if this component is of codimension b,then for somepoint x0in it,L b i∗x(L q0)survives in E∞of the latter spectral sequence.By the assumptions of the proposition we have q0+b≤d.This implies q0=d and b=0. This means that the support of L d is the whole Y.It follows that the support of K coincides with Y.The proposition is proved.Proof of the Theorem1.1.First,let us mention that ifΦE is full and faithful functor,then conditions i)and ii)are veriﬁed for obvious reasons.Indeed,it is well known fact that extension groups between skyscraper sheaves in D b coh(M)have the following form:i)Hom i X(O t1,O t2)=0for every i and t1=t2;ii)Hom i X(O t,O t)=Λi T M,t,for i∈[0,dimM],Hom i X(O t,O t)=0,for i/∈[0,dimM].Here t,t1,t2are points of M,T M,t the tangent vector space to M at t,andΛi the i–th exterior power.10Fully faithfulness of ΦE implies that the same relations are valid for imagesΦE (O t )in D b coh (X ).In what follows we prove the inverse statement.Consider composition of ΦE with its left adjoint functor Φ∗E .We are going to prove that the canonical natural transformation α:Φ∗E ◦ΦE →id is an isomorphism of functors.This is equivalent to fully faithfulness of ΦE .Indeed,for any pair ofobjects A,B ∈D b coh (M )the natural homomorphismHom(A ,B )−→Hom(ΦE A ,ΦE B )∼=Hom(Φ∗E ΦE A ,B ),is induced by α.By lemma1.2we haveΦ∗E ∼=ΨE ∨⊗π∗ωX [n ].From proposition 1.4the object K of D b coh (M ×M ),which determines Φ∗E ◦ΦE ,isK =q 13∗(q 23∗(E ∨⊗π∗ωX )⊗q 12∗E )[n ],(4)where the morphisms q 13,q 23,q 12and πare taken from the following diagramM X M q 1X X MX MM M M Mq q 12233πWe need to prove that K is quasiisomorphic to O ∆=∆∗O M ,where ∆:M −→M ×M is the diagonal embedding,because O ∆gives the identity func-tor on D b coh (M ).Let us consider a commutative diagramXj t 1t 2−→M ×X ×M f ↓q 13↓Spec k i t 1t 2−→M ×M11Here i t 1t 2is the embedding of a geometric point (t 1t 2)in M ×M ,and f :X →Spec k the corresponding ﬁbre of q 13over this point.This diagram is useful for computing the ﬁbres of K over points of M ×M .Indeed,by lemma 1.3i ∗t 1t 2K =i ∗t 1t 2q 13∗(q 23∗(E ∨⊗π∗ωX )⊗q 12∗E )[n ]==f ∗j ∗t 1t 2(q 23∗(E ∨⊗π∗ωX )⊗q 12∗E )[n ]=f ∗(j ∗t 1t 2q 23∗(E ∨⊗π∗ωX )⊗j ∗t 1t 2q 12∗E )[n ].(5)From the commutative diagramM X M M q 1X2Xj j t t 11t 2where j t 1is the embedding x →(t 1,x ),and from the deﬁnition of ΦE one obtains:j ∗t 1t 2q 12∗E =j ∗t 1E =ΦE (O t 1).(6)Analogously,j ∗t 1t 2q 12∗(E ∨⊗π∗ωX )=ΦE (O t 2)∨⊗ωX .(7)Formulas (5),(6),(7)imply isomorphisms:i ∗t 1t 2K =f ∗(ΦE (O t 1)⊗ΦE (O t 2)∨⊗ωX )[n ]==f ∗(H om (ΦE (O t 2),ΦE (O t 1))⊗ωX )[n ]=Hom(ΦE (O t 1),ΦE (O t 2))∗.(8)The last equality comes from Serre duality on X .Apply proposition 1.5to the diagonal embedding of M in M ×M .By formula(8)and assumptions of the theorem,the object K satisﬁes the hypothesis of the proposition.Therefore,K is a pure sheaf with the support at the diagonal ∆M .The natural transformation αgives rise to a sheaf homomorphism K →O ∆.It is an epimorphism,because otherwise its image would not generate the stalk of O ∆12at some point(t,t)at the diagonal.But this would imply thatΦE(O∆)has no endomorphisms(that is,the trivial object)in contradiction with assumptions of the theorem.Let F be the kernel of this morphism,i.e.there is an exact sequence of coherent sheaves on M×M:0−→F−→K−→O∆−→0(9)We have to prove that F is trivial.Considering the pull back of the short exact sequence to any point from M×M we obtain a long exact sequence showing that the sheaf F satisﬁes hypothesis of proposition1.5.It follows from the proposition that the support of F coincides with the diagonal∆M.It is suﬃcient to prove that the restriction of F to the diagonal is zero.Let us consider for this the commutative diagram:M×X−→M×X×Mp↓↓M∆−→M×Mwhere vertical morphisms are natural projections.Applying lemma1.3to the object (q23∗(E∨⊗π∗ωX)⊗q12∗E)[n]from D b coh(M×X×M)and formula(4)we obtain a formula for the derived functors of the restriction–to–diagonal functor for K:L i∆∗(K)=R n−i p∗(E⊗E∨⊗π∗ωX).Therefore,by the relative version of Serre duality and hypothesis of the theorem∆∗K=O∆,L1∆∗(K)=R1p∗(E⊗E∨)∨.Unfortunately,it is not suﬃcient to know that the restriction of K to the diagonal is O∆,because K might not be the push forward along∆of a sheaf on M(being,‘situated’on some inﬁnitesimal neighborhood of∆(M)).Furthermore,L1∆∗(O∆)=Ω1M,this means that the long exact sequence,obtained from(9)by tensoring with O∆looks as follows:···−→R1p∗(E⊗E∨)∨β−→Ω1M−→L0∆∗F−→O∆∼−→O∆−→0.13Since the support of F coincides with∆(M),so does the support of L0∆∗F, i.e.L0∆∗F is not a torsion sheaf.Therefore,if L0∆∗F is not zero,then this exact sequence shows thatβ∗:T M−→R1p∗(E⊗E∨)has a non-trivial kernel.Remark.If one considerΦE(O t)as a system of objects from D b coh(X) parametrized by M,then the restriction ofβ∗to any point t from M is,actually, the homomorphism from the deformation theoryT M,t−→Hom1X(ΦE(O t),ΦE(O t)).Therefore,a vectorﬁeld from the kernel ofβ∗gives a direction,which the objects do not change along with.This is in contradiction with the orthogonality assumptions of the theorem.Unfortunately,integrating such an algebraic vectorﬁeld one might obtain non-algebraic curves.For this reason our further strategy is going toﬁnd only a formal one–parameter deformation at one point t0in M,along which E has a formal connexion(analog of trivialization),and then to bring this in contradiction with the property ofΦE(O t)to0 having the support in point t0,which is a consequence of the orthogonality condition.Consider a point t0in M,U an open neighborhood of t0and a non-zero at t0local sectionξ∈H0(U,T M U),which belongs to the kernel ofβ∗.This vector ﬁeldξdeﬁnes a formal1–dimensional subschemeΓof the formal neighborhoodˆU of t0in M.The deﬁning ideal of the subscheme consists of the function onˆU having trivial all iterated derivatives alongξat point t0(the zero derivative being the value of a function at t0):I= f∈H0(ˆU,O) ξk(f)|t o=0,for any k≥0 .It follows that the restriction ofβ∗to the tangent bundle TΓ,t ofΓat t0is trivial.Denote EΓthe restriction of E toΓ×X.One has14Hom1Γ×X(p∗TΓ,E∨Γ⊗EΓ)∼=Hom1Γ(TΓ,p∗(E∨Γ⊗EΓ))∼=Hom0Γ(TΓ,R1p∗(E∨Γ⊗EΓ)),(10)since TΓis free(of rank1)onΓ.Let us consider theﬁrst inﬁnitesimal neighborhood∆1Γof the diagonal∆Γ:Γ×X−→Γ×Γ×X.Pulling EΓback toΓ×Γ×X along theﬁrst coordinate, then restricting to∆1Γand then pushing forward along the second coordinate,one obtains the object J1(EΓ)∈D b coh(Γ×X)of‘ﬁrst jets’of EΓ.It is included in an exact triangle:−→EΓ⊗p∗Ω1Γ[1].(11) EΓ⊗p∗Ω1Γ−→J1(EΓ)−→EΓat EHere at E is the so called Atiyah class of E.It can be considered as an element of Hom1Γ×X(p∗TΓ,E∨Γ⊗EΓ).Under identiﬁcation from(10)at E comes into the restriction ofβ∗toΓ,which is the trivial element of Hom0Γ(TΓ,R1p∗(E∨Γ⊗EΓ)) by the choice ofΓ.We consider EΓas an element of the derived category of quasicoherent sheaves on X.It is naturally endowed with an additional homomorphism A→End X EΓ,where A is an algebra of functions onΓ(isomorphic to k[[t]]).Such a homomorphism we call by A–module structure on EΓ.An A–module structure on EΓinduces A–module structures on EΓ⊗p∗Ω1Γand J1(EΓ),so that morphisms from(11)are compatible with them.Like for usual vector bundles there exists a natural homomorphism in D b Qcoh(X):EΓµ−→J1(EΓ),which is a diﬀerential operator of theﬁrst order with respect to the A–module struc-tures.Triviality of at E implies existence of a morphismJ1(EΓ)ν−→EΓ⊗p∗Ω1Γ,which is a section of theﬁrst morphism from(11).The composition∇=ν◦µdeﬁnes a morphism of quasicoherent sheaves onΓ×X∇:EΓ−→EΓ⊗p∗Ω1Γ,15which is a connexion on EΓalong theﬁbres of the projection pΓ:Γ×X−→Γin the sense that if t∈A is a function on our formal schemeΓ,then the following equality for morphisms from EΓto EΓ⊗p∗Ω1Γis valid:∇◦t−t◦∇=dt,(12)here t is identiﬁed with the corresponding morphism from EΓto EΓand dt denotes the operator of tensor multiplication by dt.SinceΓis a one-dimensional subscheme,Ω1Γis a one-dimensional free A–module. Hence,for the reason of simplicity we can identify EΓwith EΓ⊗p∗Ω1Γby means of tensoring with dt,where t∈A is a formal parameter on the schemeΓ.Then, formula(12)gives the coordinate-impulse relation between∇and t:[∇,t]=1.(13)Lemma1.6Under the above identiﬁcation of EΓwith EΓ⊗p∗Ω1Γthe morphism ∇◦t is invertible in End X EΓProof.From(13)one has:[∇,t k]=kt k−1.This gives a formula for the inverse to∇◦t:(∇◦t)−1=∞k=0(−1)k t k◦∇kProposition 1.7i)The composition λof ρ×id A with the multiplication morphism E Γ×A −→E Γ:λ:E 0×A ρ×idA −→E Γ×A −→E Γis an isomorphism,in other words it yields trivialization of E Γ.ii)E 0is quasiisomorphic to a complex of coherent sheaves on X .Proof.Let us consider the cone C of λ:E 0×A λ−→E Γ−→C.Restricting this exact triangle to the ﬁbre X 0of p Γover the closed point of Γ(which is,of course,naturally identiﬁed with X ),one obtains an exact triangle :E 0λ0−→E Γ X 0−→C X 0.Vanishing of C X 0implies vanishing of C ,hence,for proving i)we need to showthat the left morphism λ0of this triangle is isomorphism.Multiplication by t gives an exact triangle of sheaves on Γ×X :0−→O Γ×X t −→O Γ×X −→O X 0−→0It lifts to an exact triangle:E Γt −→E Γ−→E Γ X 0−→E Γ[1].Consider an octahedral diagram of exact triangles[BBD]:E E E E Gt t λ00ΓΓE ΓX 0ΓBy lemma 4.3∇◦t is an isomorphism.Hence,G is zero object and λ0is an isomorphism.Since E X 0is the restriction of complex of coherent sheaves to X 0,17E0is coherent over X0.Since C X0=G[1]is zero,so is C.Therefore,λis isomorphism.) In order toﬁnish the proof of theorem1.1let us look at the image L=Φ∗E◦ΦE(O t0 of O tunder the functorΦ∗E◦ΦE:D b coh(M)−→D b coh(M).Recall that by lemma1.2[n].Φ∗E=ΨE∨⊗π∗ωXThe trivialization of E alongΓfrom proposition1.7gives us a similar trivial-ization of E∨⊗π∗ωX.By the deﬁnition ofΨthis implies trivialization alongΓof any object from the image ofΦ∗E.Since we know thatΦ∗E◦ΦE is determined by sheaf K,having the diagonal∆(M)as its support,the image L of a skyscraper is a non–zero object from D b coh(M)having t0as the support.sheaf O tThis means that L annihilates by some power I k of the maximal ideal I⊂A. Such an object has a trivialization only if it is zero.Thisﬁnishes the proof of theorem 1.1.The simplest example of a full and faithful functor D b coh(M)−→D b coh(X)rises in the case when M is a point.In this situation we have the only object E∈D b coh(X), which is an exceptional one:Hom0X(E,E)=k,Hom i X(E,E)=0,for i=0It gives a functor from the derived category of vector spaces over k to D b coh(X).Mukai in[Mu1]and[Mu2]considered two important examples of fully faithful functors between geometric categories.First one is the so called Fourier–Mukai transform.It gives equivalenceD b coh(A)−→D b coh(ˆA)for any abelian variety A and its dualˆA.We brieﬂy recall his construction.18Let A be an abelian variety of dimension g,ˆA its dual abelian variety and P the normalized Poincare bundle on A×ˆA.AsˆA is a moduli space of invertible sheaves on A,P is a linear vector bundle,and normalization means that both P A×ˆ0 and P 0×ˆA are trivial.Theorem 1.8([Mu1]).The functorsΦP:D b coh(A)→D b coh(ˆA)andΨP: D b coh(ˆA)→D b coh(A)are equivalences of triangulated categories andΨP◦ΦP∼=(−1A)∗[g],)∗[−g],ΦP◦ΨP∼=(−1ˆAhere(−1A)∗is the auto-equivalence of D b coh(A)induced by the automorphism of multiplication by−1on A.Proof.(see[Mu1]).In the case of a principally polarized abelian variety(A,L),where L is a polar-ization,the dualˆA is identiﬁed with A.ThenΦP can be regarded as an auto-equivalence of D b coh(A).ΦP in couple with the functor of tensoring by L generates the action of the Artin braid group B3on three strands.The other example of Mukai is a K3–surface S,while M is a moduli space of stable vector bundles.Speciﬁcally,for a smooth K3–surface S one consider the Mukai lattice M(S), which is the image of the Chern homomorphism K0(S)−→H∗(S,C)from the Grothendieck group K0(S)to full cohomology group H∗(S,C).There is the Euler bilinear form on M(S),which for vectors v and v′presented by some sheaves F and F′is deﬁned by the formula:χ(v,v′)= (−1)i dimExt i(F,F′).Since the canonical class is trivial,by Serre duality this form is symmetric.Let v be an isotropic indivisible by integer vector with respect toχ.The coarse moduli space of stable bundles on S,corresponding to v,is again a smooth K3–surface S′.There is a rational correspondence between S and S′.If S′is aﬁne moduli space,then we have the universal vector bundle E on S×S′.19Theorem1.9[Mu2].FunctorΦE:D b coh(S)−→D b coh(S′)is an equivalence of triangulated categories.In the both examples of equivalences the canonical class of varieties(either of abelian one or of a K3–surface)is trivial.In chapter3we construct another example of equivalence between geometric categories usingﬂops.The centre of such transfor-mation is in a sense trivial with respect to the canonical class.An explanation for this phenomenon is given in chapter4.2Intersection of two even dimensional quadrics.In this chapter we show how theorem1.1helps to construct a semiorthogonal decom-position of the derived category of coherent sheaves on the intersection of two even dimensional quadrics,with one summand being the derived category on a hyperelliptic curve and with the others being generated by single exceptional objects.This result can be considered as a categorical explanation for the description,due to Desale and Ramanan,of moduli spaces of rank2vector bundles on a hyperelliptic curve as a base of a family of projective subspaces belonging to the intersection of two even dimensional quadrics[DR].Our construction gives analogous description for any moduli spaces of bundles on the curve by means of families of complexes of coherent sheaves on the intersection locus.Weﬁrst recall some deﬁnitions and facts concerning exceptional sequences,admis-sible subcategories,Serre functors and semiorthogonal decompositions[Bo],[BK].Let B be a full subcategory of an additive category.The right orthogonal to B is the full subcategory B⊥⊂A consisting of the objects C such that Hom(B,C)=0 for all B∈B.The left orthogonal⊥B is deﬁned analogously.If B is a triangulated subcategory of a triangulated category A,then⊥B and B⊥are also triangulated subcategories.Deﬁnition2.1Let B be a strictly full triangulated subcategory of a triangulated category A.We say that B is right admissible(resp.,left admissible)if for each X∈A there is an exact triangle B→X→C,where B∈B and C∈B⊥(resp.,20。

Duality and the Legendre Transform

∂2F (A) ∂A2 = −
∂2FD(AD) ∂A2D
−1
,
(4)
and ImF (z) is a convex function. Using (3) and (4) it is easy to see that an equivalent deﬁnition of ΓD(AD) is given by
arXiv:hep-th/9508140v1 25 Aug 1995
Abstract
We deﬁne a weak-strong coupling transformation based on the Legendre transformation of the eﬀective action. In the case of N = 2 supersymmetric Yang-Mills theory, this coincides with the duality transform on the low energy eﬀective action considered by Seiberg and Witten. This Legendre transform interpretation of duality generalizes directly to the full eﬀective action, and in principle to other theories.
1In order to deﬁne the eﬀective action, Γ[Aa], one must impose a constraint on J a (which matches the Bianchi constraint on Aa) chosen so that δW[J b]/δJ a = Aa is invertible.

Quantization of soliton systems and Langlands duality

QUANTIZATION OF SOLITON SYSTEMS AND LANGLANDS DUALITY
arXiv:0705.2486v3 [math.QA] 9 Dec 2007
BORIS FEIGIN AND EDWARD FRENKEL Abstract. We consider the problem of quantization of classical soliton integrable systems, such as the KdV hierarchy, in the framework of a general formalism of Gaudin models associated to aﬃne Kac–Moody algebras. Our experience with the Gaudin models associated to ﬁnite-dimensional simple Lie algebras suggests that the common eigenvalues of the mutually commuting quantum Hamiltonians in a model associated to an aﬃne algebra b g should be encoded by aﬃne opers associated to the Langlands dual aﬃne algebra Lb g. This leads us to some concrete predictions for the spectra of the quantum Hamiltonians of the soliton systems. In particular, for the KdV system the corresponding aﬃne opers may be expressed as Schr¨ odinger operators with spectral parameter, and our predictions in this case match those recently made by Bazhanov, Lukyanov and Zamolodchikov. This suggests that this the correspondence between quantum integrals of motion and diﬀerential operators may be viewed as special cases of the Langlands duality.

Ergodic solenoidal homology

ERGODIC SOLENOIDAL HOMOLOGY
arXiv:math/0702501v1 [math.DG] 16 Feb 2007
˜ ´ VICENTE MUNOZ AND RICARDO PEREZ MARCO Abstract. We deﬁne generalized currents associated with immersions of abstract solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De Rham’s realization of the real homology by only using a restricted geometric subclass of currents. These generalized currents do extend Ruelle-Sullivan and Schwartzman currents. We extend Schwartzman theory beyond dimension 1 and provide a uniﬁed treatment of Ruelle-Sullivan and Schwartzman theories via Birkhoﬀ’s ergodic theorem for the class of immersions of controlled solenoids. We develop some intersection theory of these new generalized currents that explains why the realization theorem cannot be achieved only with Ruelle-Sullivan currents.

Extensions of some results of P. Humbert on Bezout’s identity for

Journal of Computational and Applied Mathematics 196(2006)212–228/locate/camExtensions of some results of P.Humbert on Bezout’s identity forclassical orthogonal polynomialsI.Area a ,E.Godoy b ,∗,A.Ronveaux c ,A.Zarzo da Departamento de Matemática Aplicada II,E.T.S.E.Telecomunicación,Universidade de Vigo,36200-Vigo,Spainb Departamento de Matemática Aplicada II,E.T.S.Ingenieros Industriales,Universidad de Vigo,36200-Vigo,Spainc Département de Mathématique,Unitéd’Analyse Mathématique et de Mécanique,UniversitéCatholique de Louvain,Bâtiment Marc de Hemptinne,Chemin du Cyclotron 2,B-1348Louvain-la-Neuve,Belgiumd Instituto Carlos I de Física Teórica y Computacional,Facultad de Ciencias,Universidad de Granada,Spain and Departamento de MatemáticaAplicada,E.T.S.Ingenieros Industriales,Universidad Politécnica de Madrid,SpainReceived 28January 2005AbstractIn this paper,the Bezout’s identity is analyzed in the context of classical orthogonal polynomials solution of a second order differential equation of hypergeometric type.Differential equations,relation with the starting family as well as recurrence relations and explicit representations are given for the Bezout’s pair.Extensions to classical orthogonal polynomials of a discrete variable and their q -analogues are also presented.Applications of these results for the representation of the second kind functions are given.©2005Elsevier B.V .All rights reserved.MSC:Primary 33C25;secondary 33C20;05A10Keywords:Orthogonal polynomials;Bezout identity;Second kind functions1.IntroductionThe well-known Bezout’s identity [8],which play a relevant role in several applied problems such as system design and analysis [26]or linear time-varying ordinary differential control systems [24],can be stated in the following way:if P (x)and Q(x)are two polynomials,then there exist two other polynomials A(x)and B(x)such thatA(x)P (x)+B(x)Q(x)=gcd (P ,Q),where gcd (P ,Q)is the greatest common divisor of P (x)and Q(x),which becomes a non-zero constant (usually ﬁxed to 1)when P (x)and Q(x)are relatively prime.With no other assumptions on the polynomials P (x)and Q(x),there exists an inﬁnite number of Bezout pairs (A,B)corresponding to these two polynomials.However,if we assume that the degree of A(x)is smaller than the degree of Q(x)and the degree of B(x)is smaller than the degree of P (x),then the Bezout pair is unique [28].In this situation,∗Corresponding author.Tel.:+34986812157;fax:+34986812116.E-mail addresses:area@dma.uvigo.es (I.Area),egodoy@dma.uvigo.es (E.Godoy),ronveaux@math.ucl.ac.be (A.Ronveaux),azarzo@etsii.upm.es (A.Zarzo).0377-0427/$-see front matter ©2005Elsevier B.V .All rights reserved.doi:10.1016/j.cam.2005.09.002I.Area et al./Journal of Computational and Applied Mathematics196(2006)212–228213 A(x)and B(x)can be computed easily from the Euclidean division of P(x)by Q(x),which becomes a heavy problem if the degree of P(x)and Q(x)are large enough.This can be done by using PARI/GP computer algebra system[23], originally developed by Henri Cohen and his co-workers.In this paper,we propose an alternative(and explicit)way to obtain A(x)and B(x)with the following assumptions on P(x)and Q(x):we impose that P(x)=P n(x)is a polynomial of degree n belonging to a family of classical orthogonal polynomials,continuous[9],discrete[22]or their q-analogues[19];also Q(x)=D P(x),where D is the derivative operator d/d x,the backward difference operator∇or the q−1-difference operator D q−1,respectively.This problem was partly treated by Humbert[14,15]in the case of classical continuous orthogonal polynomials,andA n(x)P n(x)+B n(x)P n(x)= n,where n is a constant.In these two papers[14,15],with the same title and very different results,Humbert studied in detail the Legendre family,giving also some properties in Gegenbauer,Laguerre,Hermite and Jacobi cases.As it is indicated in[16]these identities are useful in rotatingﬂuids problems,when computing the Legendre’s function of second kind(see Section5).At the Humbert’s period(1915–1920),link between three-term recurrence relation and orthogonal polynomials were not very popular,even known.Of course,these papers of Humbert[13–16]do not mention the orthogonality of the Bezout’s polynomials A n(x)and B n(x),focussing his interest in representations of the second kind Legendre functions.Despite the fact that Humbert presented most of the results on the Legendre case,and also some properties for Hermite,Laguerre,Gegenbauer and Jacobi polynomials,80years later it seems more appropriate to give a general presentation valid for classical continuous polynomials,for classical orthogonal polynomials of a discrete variable and also for their q-analogues.The structure of the paper is the following:In Section2,we study the classical continuous situation.Differential equations for the polynomials A n(x)and B n(x)are obtained,assuming that{P n(x)}is a family of classical orthogonal polynomials[22].Moreover,we present some new results linking explicitly the polynomials A n(x)and B n(x)with the derivative of the starting family P n(x)and theﬁrst associated polynomials of the starting family P(1)n(x).Recurrence relation and orthogonality property for the family{A n(x)}are also presented.The sequence{B n(x)}satisﬁes also a three-term recurrence relation.In Sections3and4,the results are extended(in general without proofs)to the cases of classical orthogonal polynomials of a discrete variable and their q-analogues,respectively.Finally,in Section5,some applications of these results are presented to obtain a representation of the second kind solutions of the q-difference equation of hypergeometric type.2.Classical continuous caseThe hypergeometric type differential equation is a second-order homogeneous differential equation L2[y(x)]:= (x)y (x)+ (x)y (x)+ n y(x)=0,(1) where (x)and (x)are polynomials of degree at most two and one,respectively,and n is a constant.The above differential equation has a unique monic polynomial solution P n(x)of degree exactly n if and only ifn=−n+(n−1)2(2)and n= m for m=0,1,2,...,n−1.Under some assumptions[9,22],it is possible toﬁnd a positive weight function (x)and a interval(a,b)such that( (x) (x)) = (x) (x), (x) (x)x |x=bx=a=0, =0,1,2,....In these conditions,the polynomial solutions P n(x)of(1)for = n are the classical orthogonal polynomials of Jacobi,Laguerre,and Hermite and satisfy the orthogonality relationbaP n(x)P m(x) (x)d x=d2n n,m,214I.Area et al./Journal of Computational and Applied Mathematics196(2006)212–228where n,m denotes the Kronecker’s delta.Assuming that P n(x)are monic,these families satisfy also a three-term recurrence relationxP n(x)=P n+1(x)+ n P n(x)+ n P n−1(x),n 1,P0(x)=1,P1(x)=x− 0,(3) where n and n>0can be expressed(see[12]and the references therein)in terms of the polynomials (x)and (x) of(1)asn=1n−1 n[ n(1−n)(0) − 2n(0)],n 0, −1=0,(4)n=n n/2−12( n−1) n−3/2 n−1/2{2 [ (0) n−1(0)− (0) ]− [ n−1(0) 1−n(0)+4(n−1) (0) (n−1)/2]},n 0, −1=0.(5)Here the notation (x)= (x)+ (x)has been used and will also used throughout this section.As it is well-known[27],in these conditions P n(x)has exactly n real and different roots,which interlace with those of P n(x),for n 2.Our interest is to study,in this context,properties of the polynomials A n(x)and B n(x)(Bezout’s pair),which existence is guaranteed by the Bezout’s identity,satisfyingA n(x)P n(x)+B n(x)P n(x)= n( 1=1),(6) where deg(A n(x))<deg(P n(x))=n−1and deg(B n(x))<deg(P n(x))=n.For n=0and n=1,we obtain that A0(x)=1,A1(x)=0,B0(x)=0and B1(x)=1.In this latter equation,the constants n are introduced allowing us to re-normalize the polynomials A n(x)and B n(x)appropriately,which is useful when recovering some criteria on the multiplicity of the zeros of P n(x).2.1.Differential equationIn this section,we obtain a second-order differential equation for the polynomials A n(x)and for the polynomials B n(x).Relations(6)and(1)generate by derivation the following equivalent relations:(x)A n(x)− n B n(x)=−P n(x)N(x),(7) (x)A n(x)+ (x)B n(x)− (x)B n(x)=P n(x)N(x),(8) where,atﬁrst,N(x)is a polynomial,but the compatibility in the degree implies that N(x)=N becomes a constant and also that A n(x)is of degree exactly equal to n−2,and B n(x)is of degree exactly equal to n−1.By elimination and derivation in(7)and(8),we get the following non-homogeneous second order differential equations for the polynomials A n(x),(x)A n(x)+( (x)− (x))A n(x)+ n A n(x)=−2NP n(x),(9) and for the polynomials B n(x),(x)B n(x)− (x)B n(x)+n− (x)+(x) (x)(x)B n(x)=2NP n(x)−N (x)(x)P n(x).(10)Note again that these differential equations coincide with the ones obtained by Humbert[15]in the Legendre case ( (x)= (x)).On the other hand,a choice of a constant N independent of the degree n shall beﬁxed in Proposition2.1and shall precise the normalization for the polynomials A n(x)and B n(x).I.Area et al./Journal of Computational and Applied Mathematics196(2006)212–2282152.2.Identiﬁcation of the Bezout’s pairIn this section we identify the polynomials A n(x)and B n(x)of(6)in relation with the starting classical orthogonal family P n(x).Representation and recurrence relation for the polynomials A n(x),of degree n−2,given below ensures their orthogonality for n 2.The corresponding orthogonality weight are particular cases of the general associated orthogonal polynomial weight given in[6,29].Explicit representation and recurrence relation for the B n(x)polynomials will also be given.Moreover,an example related to Jacobi polynomials is presented.Proposition2.1.For n 2,A n(x)=[P n(x)](1),(11) where[P n(x)](1)denotes theﬁrst associated polynomial of theﬁrst derivative P n(x)of P n(x).Proof.Theﬁrst associated monic polynomial P(1)n−1(x)of the orthogonal family P n(x)of weight (x)is deﬁned asP(1)n−1(x)=1M0baP n(x)−P n(s)x−s(s)d s,where(a,b)is the orthogonality interval andM0= ba(s)d s.It is well known that for classical families[25]L∗2[P(1)n−1(x)]=( −2 )P n(x),where the operator L∗2deﬁned byL∗2[y(x)]= (x)y (x)+(2 (x)− (x))y (x)+( n+ − )y(x),is the formal adjoint of L2introduced in(1).Moreover,theﬁrst derivatives P n(x)=z(x)are solution of[22]L2[z(x)]= (x)z (x)+( (x)+ (x))z (x)+( n+ )z(x)=0.Therefore,theﬁrst associated polynomial of theﬁrst derivatives of P n(x),w(x)=[P n(x)](1),are solution of [L2]∗[w(x)]= (x)w (x)+( (x)− (x))w (x)+ n w(x)=(−2 − )P n(x).(12) With A n(x)=K[P n(x)](1)=Kw(x)we obtain from(9)and(12)[L2]∗[A n(x)/K]=(−2 − )P n(x)=−2NP n(x)/K,and then K=2N/(2 + ).Therefore,A n(x)=2N2 +[P n(x)](1).By choosing K=1,so that the constant N is(see Eq.(2)):N=(2 + )2=−22,(13)we obtain that the differential equation satisﬁed by the polynomials A n(x)is(x)A n(x)+( (x)− (x))A n(x)+ n A n(x)= 2P n(x),(14)216I.Area et al./Journal of Computational and Applied Mathematics196(2006)212–228andA n(x)=[P n(x)](1),n 2.As a consequence of the above proposition,we haveCorollary2.2.The three-term recurrence relation satisﬁed by the polynomials A n(x)isxA n(x)= n A n+1(x)+ n A n(x)+ n A n−1(x),(15) valid for n 2with the initial conditionsA0(x)=1,A1(x)=0,A2(x)=2,wheren=nn+1, n= n−F n, n= n−G n,n and n are given in(4)and(5),respectively and,with the already introduced notation (x)= (x)+ (x),F n= (0) − (0)n−1n,G n=(n−1)/2− (2n−1)/22(2n−3)/2( n−1)2 (2n−1)/2{ n−1(0)[2 (0) − 1−n(0) ]−2 (0)( n−1)2}.Next,we give the representation and recurrence relation for the B n(x)family. Proposition2.3.The polynomials B n(x)are given byB n(x)=2 (x){[P n(x)](1)} − 2P n(x)2 n,n 2.Proof.Straightforward consequence of(7)by using that N=− 2/2,from the choice K=1in Proposition2.1. Since B k(x)is a polynomial of degree exactly k−1,the expansion of xB n(x)in terms of B j(x),j=1,...,n+1 is unique.From easy computations we obtain the following result generalizing the results obtained by Humbert[15] in the particular case of Gegenbauer.Proposition2.4.The polynomials B n(x)satisfy the three-term recurrence relationB0(x)=0,B1(x)=1,xB n(x)=ˆ n B n+1(x)+ˆ n B n(x)+ˆ n B n−1(x),n 1,(16) whereˆ n=−1,n=1,1,n 2,ˆn=+ 12,n=1,n,n 2,ˆ n=22,n=2,n,n=2,and n and n are given by(4)and(5)respectively.Remark1.Although it is evident from(16),we would like to emphasize that the B n(x)family satisfy,for n 2,the same recurrence formula as the starting P n classical family(see(3)),but with the initial conditions B0(x)=0,B1(x)=1 and B2(x)=−P 2(x)/2.Proposition2.5.For n 2,we haveB n(x)= (c1(x)P n(x)− (x)P(1)n−1(x)),(17)I.Area et al./Journal of Computational and Applied Mathematics196(2006)212–228217 where is a constant given by=− 22[ (0)( )+ (0)(( /2) (0)− (0) )],(18)andc1(x)=2x+(0) −( /2) (0).(19)Proof.The polynomials B n(x)being solution of(16)like P n(x)and P(1)n−1(x),(cf.relation3),there exist two(unique) polynomials in x and independent of n,let us call R(x)and S(x)such thatB n(x)=R(x)P n(x)+S(x)P(1)n−1(x),n 3.(20) The polynomials R(x)and S(x)can be computed from the explicit representation of B n(x)given in Proposition2.3, and A n(x)given in Proposition2.1by equating the coefﬁcients in the powers of x using Eqs.(4)and(5). Finally,in the classical continuous case we have the following general result.Proposition2.6.For classical continuous orthogonal polynomials,the Bezout’s identity(6)is given byA n(x)P n(x)+B n(x)P n(x)= n,n 1,(21) where A n(x)and B n(x)are given in Propositions2.1and2.3,respectively, 1=1,2=4 ( (0) − (0)( (0)+ (0)))+2( (0)2− (0)2+4 (0) ) +4 (0)( )2( + )2(2 + ),and for n 3,n=−n[(2n−1)!]22 nj=12n+j−1n−4j=03+jn−1j=1[(4j2 (0)( /2)2+ /2(( (0))2−(j (0))2+4j (0) )+ ( (0) − j(0) (0)))].(22) Here,the value of the constant n is a consequence of the choice of the constant N made in the proof of Proposition 2.1(see Eq.(13)).2.3.Example:Jacobi caseMonic Jacobi polynomials P( , )n(x)are solution of(1)with[27](x)=1−x2, (x)= − −( + +2)x.For n 2,with P n(x)=P( , )n(x),A n(x)P n(x)+B n(x)P n(x)= n=−n!4n−1( +2)n−1( +2)n−1 ( + +n+1)n( + +4)2n−3,where the constant n is never zero inside the classical orthogonality region( >−1and >−1)for Jacobi polynomials. Moreover, n=0if and only if P( , )n(x)has multiple roots,which can be located at x=−1or x=1.The multiplicity of this zero is| |or| |such that( +2)n−1or( +2)n−1is zero.The above result is in accordance with[27,p.145], where a distribution of the zeros of the general Jacobi polynomials is presented.218I.Area et al./Journal of Computational and Applied Mathematics196(2006)212–228 Moreover,the three-term recurrence relation(16)for the B n(x)polynomials can be written as xB n(x)=ˆ n B n+1(x)+ˆ n B n(x)+ˆ n B n−1(x),n 1,B0(x)=0,B1(x)=1, whereˆ n=−1,n=1,1,n 2,ˆn=−+ +4,n=1,n,n 2,ˆ n=⎧⎨⎩−8( +2)( +2)( + +4)2( + +5),n=2, n,n=2,and n and n are the coefﬁcients of the three-term recurrence relation(3)satisﬁed by monic Jacobi polynomials. Therefore,in this Jacobi case,the above results prove the orthogonality of the family B n(x),of degree n−1,but with respect to an unknown weight.3.Extension to difference equationsClassical discrete orthogonal polynomials P n(x)(Hahn,Kravchuk,Meixner and Charlier)are solution of a second-order difference equation similar to(1)D2[y(x)]= (x) ∇y(x)+ (x) y(x)+ n y(x)=0,(23) where f(x)=f(x+1)−f(x)and∇f(x)= f(x−1).From the interlacing property of the two families y(x)=P n(x) and P n(x)(or∇P n(x))[1],the search for Bezout’s pair is still relevantA n(x)P n(x)+B n(x)∇P n(x)= n,where n is a constant.3.1.Difference equationsDifference equation for A n(x)is obtained in a similar way,using now the difference equation(23)satisﬁed by P n(x). From(x) (A n(x)P n(x)+B n(x)∇P n(x))=0,we get(x) A n(x)− n B n(x)=− P n(x)N(x),(24) (x)(A n(x)+ B n(x)+ A n(x))− (x)B n(x)=P n(x)N(x),(25) where N(x)is a polynomial.The compatibility in the degree implies that N(x)=N is a constant.In this case,we obtain the following difference equation for the polynomials A n(x)[ (x−1)+ (x−1)] ∇A n(x)+[ (x)− (x−1)− (x−1)] A n(x)+ n A n(x)=−N[ ∇P n(x)+ ∇P n(x−1)]=−N[ +∇]∇P n(x).Similarly,a second order difference equation for the polynomial B n(x)could be given.3.2.Identiﬁcation of the Bezout’s pairAs in the continuous case,N=− 2/2and we have the following result linking the polynomials A n(x)with theﬁrst associated polynomials of P n(x).I.Area et al./Journal of Computational and Applied Mathematics196(2006)212–228219 Proposition3.1.The polynomials A n(x)and P n(x)in(23)are related byA n(x)=[∇P n(x)](1),n 2,where[∇P n(x)](1)denotes theﬁrst associated polynomial[7]of the difference derivative∇P n(x)of P n(x). Mutatis mutandis,classical discrete orthogonal polynomials also satisfy a number of properties similar as in Section 2.Assuming that P n(x)are monic,they satisfy a three-term recurrence relationxP n(x)=P n+1(x)+ n P n(x)+ n P n−1(x),n 1,P0(x)=1,P1(x)=x− 0,(26) wheren=− (0)( − )−n( +2 (0))( +(n−1)/2 )((n−1) + )(n + ),(27)n=−n( +(n−2)/2 )( +(2n−3)/2 )( +(2n−1)/2 )[ ( n−1)+ ( n−1)],(28) andn=− (0)+n (0)−n2 /2+n .Also,we have the following difference representation[3]P n(x)= P n+1(x)n+1+H n P n(x)+J n P n−1(x),whereH n=( n)2n( +(n−1) /2)−12,J n=−n2 +(n−2).As in the continuous case,Proposition3.2.The polynomials A n(x)satisfy the following three-term recurrence relationxA n(x)= n A n+1(x)+ n A n(x)+ n A n−1(x),(29) valid for n 2with the initial conditionsA0(x)=1,A1(x)=0,A2(x)=2,wheren=nn+1, n= n−H n, n= n−J n.Next,we give a relation between the polynomials B n(x)in terms of the starting family P n(x)as well as a three-term recurrence relation satisﬁed by B n(x).Proposition3.3.For n 2,the polynomials B n(x)can be computed from A n(x)and P n(x)by usingB n(x)=2 (x) {[∇P n(x)](1)}− 2 P n(x)2 n.(30)Proposition3.4.The polynomials B n(x)satisfy the three-term recurrence relationB0(x)=0,B1(x)=1,xB n(x)=ˆ n B n+1(x)+ˆ n B n(x)+ˆ n B n−1(x),n 1,(31)220I.Area et al./Journal of Computational and Applied Mathematics 196(2006)212–228whereˆ n =−1,n =1,1,n 2,ˆ n = 0+ 1−12,n =1, n ,n 2,ˆ n =22,n =2, n ,n =2,where n and n are given in (27)and (28).Remark 2.Although it is evident from (31),we would like to emphasize that,in this discrete case,the B n family also satisfy,for n 2,the same recurrence formula as the starting P n discrete family (see (26)),but with the initial conditions B 0=0,B 1(x)=1and B 2(x)=− P 2(x)/2.Proposition 3.5.For n 2,we haveB n (x)= (c 1(x)P n (x)− (x)P (1)n −1(x)),where and c 1(x)are deﬁned in (18)and (19),respectively .For classical discrete orthogonal polynomials (Hahn,Kravchuk,Meixner and Charlier) (0)=0and we have the following general result.Proposition 3.6.If {P n (x)}is a family of classical orthogonal polynomials of a discrete variable ,we haveA n (x)P n (x)+B n (x)∇P n (x)= n ,n 1,where A n (x)and B n (x)are given in Propositions 3.1and 3.4,respectively , 1=1,2=(2( (0)+ (0)+ )+ )(2 (0) −(2 (0)− )(2 + ))2( + )2(2 + ),and for n 3,n =−n [(2n −1)!]2( /2+ (0)+ + (0))(( /2− (0))( /2+ )+ /2 (0))2 n j =1 2n +j −1 n −4j =0 3+j×n −1 j =1j 2 2+j ( (0)+ (0))+ (0) j 2− (0) j2+ + 2(0) .3.3.Example:Hahn case Monic Hahn polynomials h ( , )n(x ;M)are solution of (23)with [22](x)=x(M + −x), (x)=( +1)(M −1)−( + +2)x .In this situationA n (x)P n (x)+B n (x)∇P n (x)=−n !(M −n)!( + +M +2)n −1( +2)n −1( +2)n −1( + +4)2n −3( + +n +1)n.Note that n =0inside the orthogonality region for Hahn polynomials ( >−1, >−1and n M −1∈N ).Moreover, n =0if and only if h ( , )n (x ;M)and ∇h ( , )n (x ;M)have common roots.Furthermore,the three-term recurrence relation (31)for the B n (x)polynomials can be written asxB n (x)=ˆ n B n +1(x)+ˆ n B n (x)+ˆ n B n −1(x),n 1,B 0(x)=0,B 1(x)=1,I.Area et al./Journal of Computational and Applied Mathematics 196(2006)212–228221whereˆ n =−1,n =1,1,n 2,ˆn = (2+ )(−2+M)4+ +,n =1,n ,n 2,ˆ n = −2(2+ )(2+ )(M −2)(2+ + +M)(4+ + )2(5+ + ),n =2,n ,n =2,and n and n are the coefﬁcients of the three-term recurrence relation (26)satisﬁed by monic Hahn polynomials.Therefore,in this Hahn case,the above results prove the orthogonality of the family B n (x),of degree n −1,but with respect to an unknown weight.4.Extension to q -difference equationsLet 0<q <1.The q -classical orthogonal polynomials P n (x ;q)=P n (x)[19]are solution of a second-order q -difference equation similar to (1)D 2,q [y(x)]= (x)D q D q −1y(x)+ (x)D q y(x)+ n y(x)=0,(32)where (x)and (x)are polynomials in x of degree at most two and one in x ,respectively,which could also depend on the parameter q ,and the q -derivative operator D q is deﬁned byD q y(x)=y(qx)−y(x)(q −1)x,x =0,with D q y(0):=y (0)provided f (0)exists [19].In the above q -difference equationn =[n ]q12[1−n ]q − , =d (x)d x ,=d 2 (x)d x 2,where the q -numbers are deﬁned by[n ]q :=q n −1q −1.The notation used in this section is that of [11,Chapter 1](see also [19]).Assuming that P n (x)are monic,the q -classical orthogonal polynomials also satisfy a three-term recurrence relation [20],xP n (x)=P n +1(x)+ n P n (x)+ n P n −1(x),n 1,P 0(x)=1,P 1(x)=x − 0,(33)where by deﬁning = (0)(1+q)+(q −1) (0)and q,n =q(q n +[n ]q /2),we haven =q n2(q−1)2 q,2n q,2n −2[(2(1−q)q n ( q n −(1+q) (0))−( (q +q 2n )+q n(1+q)((−1+q)q (0)+(1+q) (0))) )],(34)n =(1−q n )q n −3(1−q) q,2n −1( q,2n −2) q,2n −3q n −q 2 2(4q 4n 2 (0)−4q 1+3n(0)−2q 3+n (0) +q 4 (0)( )2−2q 2+2n (4 2 (0)−2(−1+q)(2 (0)+ (0)2) +(−2 2+(q −1) (0)((q −1) (0)+2 (0))) )),(35)and=(q −1) (0)+(0),=(q −1)+2.Also,we have the following q-derivative representation[4,20]P n(x)=D q P n+1(x)[n+1]q+K n D q P n(x)+M n D q P n−1(x),whereK n=q n(( /2) q+ q n( q n− (0)(1+q)))(q−1) q,2n q,2n−2,(36)M n=[n]q q2n−3(q−1)2 q,2n−3( q,2n−2)2 q,2n−1(0)2q4+ 2q4n− (0)2q n+3+ q3n+1+q2n+2( (0))2 +2( 2−2 (0) ).(37)From the interlacing property of the two families y(x)=P n(x)and D q P n(x)(or D q−1P n(x))[1],the search for Bezout’s pair is relevant tooA n(x)P n(x)+B n(x)D q−1P n(x)= n.(38)4.1.q-difference equationsThe q-difference equation for A n(x)is obtained in a similar way,using now the q-difference equation(32)satisﬁed by P n(x).From(x)D q(A n(x)P n(x)+B n(x)D q−1P n(x))=0,we get(x)D q A n(x)− n B n(x)=−N(x)D q P n(x),(39) (x)(A n(x)+(q−1)xD q A n(x)+D q B n(x))− (x)B n(x)=P n(x)N(x),(40)where N(x)is a polynomial.The compatibility in the degree implies that N(x)=N is a constant.In this case,we obtain the following q-difference equation for the polynomials A n(x)[q (x)+x(1−q)D q−1 (x)+(q−1)x (x/q)]D q D q−1A n(x)+[D q−1 (x)− (x/q)]D q A n(x)+ n A n(x)=−N[qD q D q−1P n(x)+D q D q−1P n(x/q)]=−Nq[D q+D q−1]D q−1P n(x).Now,N=− 2/[2]q and,similarly to the previous cases,a second order q-difference equation for the polynomial B n(x)could be given.4.2.Identiﬁcation of the Bezout’s pairThe following propositions are easily proved in a similar way as in Section2.Proposition4.1.The polynomials A n(x)and P n(x)in(32)are related byA n(x)=[D q−1P n(x)](1),n 2,where[D q−1P n(x)](1)denotes theﬁrst associated polynomial[10]of the q−1-difference derivative D q−1P n(x)of P n(x). Proposition4.2.The polynomials A n(x)satisfy the following three-term recurrence relationxA n(x)= n A n+1(x)+ n A n(x)+ n A n−1(x),(41) valid for n 2with the initial conditionsA0(x)=1,A1(x)=0,A2(x)=[2]q q,wheren=q[n]q[n+1]q, n= n−K n, n= n−M n,and the coefﬁcients n, n,K n and M n are given in(34),(35),(36),and(37),respectively. Next,we present the results corresponding to the polynomials B n(x)in this q-case. Proposition4.3.The polynomials B n(x)can be computed from A n(x)and P n(x)by usingB n(x)=[2]q (x)D q{[D q−1P n(x)](1)}− 2D q P n(x)[2]q n,n 2.(42)Proposition4.4.The polynomials B n(x)satisfy the three-term recurrence relationB0(x)=0,B1(x)=1,xB n(x)=ˆ n B n+1(x)+ˆ n B n(x)+ˆ n B n−1(x),n 1,(43) whereˆ n=−1,n=1,1,n 2,ˆn=⎧⎨⎩0+ 1[2]q,n=1,n,n 2,ˆn=⎧⎨⎩q,1 2q2D q,n=2,n,n=2,and n and n are given in(34)and(35),respectively.Remark3.Once more from(16),we would like to emphasize(see(43))that the B n family satisfy,for n 2,the same recurrence formula as the starting P n classical family(see(33)),but with the initial conditions B0=0,B1(x)=1and B2(x)=−D q P2(x)/[2]q.Proposition4.5.For n 2,we haveB n(x)= q(c1(x)P n(x)− (x)P(1)n−1(x)),whereq=(q + /2)q( (0)( )2+ (0)(( /2) (0)− (0) )),and c1(x)is given in(19).For q-classical orthogonal polynomials we have the following general result.Proposition 4.6.Let {P n (x)}be a family of q-classical orthogonal polynomials .Then ,A n (x)P n (x)+B n (x)D q −1P n (x)= n ,n 1,where A n (x)and B n (x)are given in Propositions 4.1and 4.3,respectively , 1=1,2=2(1+q)q(2q + )(2q 2 +(1+q) )2(4q 4 (0) 2+4q 2 (− (0)(q (0)+ (0))+(1+q) (0) )+ (2q( (0)− (0))(q (0)+ (0))+(1+q)2 (0) )),and for n 3,n =−[n ]q (q ;q)22n −1(1+q)(1−q)2(2n −1)q 3n(n −1) nj =1 2n +j −1 n −4j =0 3+j ×q 4 (0) 2+q 2 (−( (0)(q (0)+ (0)))+(1+q) (0) )+ (2q( (0)− (0))(q (0)+ (0))+(1+q)2 (0) )4×n −1 j =1 2(0)[2j ]2q+q 2j(0)q 2j −((0))2[j ]q − (0)q j(0)+2q j ( (0)((1−q) (0)− (0))[j ]2q +q j (2 (0)[2j ]q +( (0))2)).(44)4.3.Example:big q-Jacobi caseBig q -Jacobi polynomials P n (x ;a,b,c ;q)are solution of (32)with [19](x)=(aq −x)(cq −x), (x)=cq −x +aq(1−(b +c)q +bqx)q −1.In this case,we obtain for n 2,A n (x)P n (x)+B n (x)D q −1P n (x)= n =(−1)n q n(n −3)/2(aq 2;q)n −1(bq 2;q)n −1(cq 2;q)n −1(abq 2/c ;q)n −1c n −1(1−q)(abq 4;q)2n −3(abq n +1;q)n,where(a ;q)0:=1,(a ;q)n =n −1 j =0(1−aq j ),n 1.Note that n =0inside the orthogonality region for big q -Jacobi polynomials (0<a <q −1,0<b <q −1,c <0).Moreover, n =0if and only if P n (x)and D −1q P n(x)have common roots.Furthermore,the three-term recurrence relation for the B n (x)polynomials can be written asxB n (x)=ˆ n B n +1(x)+ˆ n B n (x)+ˆ n B n −1(x),n 1,B 0(x)=0,B 1(x)=1,whereˆ n=−1,n=1,1,n 2,ˆn=⎧⎨⎩(a+c)q−a(b+c)q31−abq4,n=1,n,n 2,ˆn=⎧⎨⎩−a(1−q)q2(1+q)(1−aq2)(1−bq2)(abq2−c)(1−cq2)(1−abq4)2(1−abq5),n=2, n,n=2,and n and n are the coefﬁcients of the three-term recurrence relation(33)satisﬁed by monic big q-Jacobi polynomials. Therefore,in this big q-Jacobi case,the above results prove the orthogonality of the family B n(x),of degree n−1,but with respect to an unknown weight.5.ApplicationsIn this section,we give a representation of the second kind solutions of(32)in terms of the polynomials B n(x)as well as the q-classical polynomials P n(x).The second kind non polynomial solution u2(x;q)=u2(x)of(32),which is linearly independent of P n(x),can be obtained by using the q-analogue of the variation of constants technique as[5]u2(x)=P n(x) Cq,n(x)P n(x)P n(qx)d q x,where d q x is the Fermat measure[2,18]and C q,n(x)is the q-Casoratian deﬁned byC q,n(x)=C q(u1(x),u2(x))=u1(x)u2(x)D q(u1(x))D q(u2(x)).The q-Casoratian is given by[5]C q,n(x)=⎧⎪⎨⎪⎩1,n=0, (q;q)nn−1j=0([2+j−2n]q /2− )(1−q)n (qx) (qx),n 1,which with an appropriate normalization of the polynomials P n(x),is always different from zero if[2+k−2n]q2− =0,k=0,...,n−1,(45)holds true.Note thatℵn=(q;q)n(1−q)n n−1j=0[2+j−2n]q2−is up to a non-vanishing constant the coefﬁcient A k,n appearing in the Rodrigues formula for the k th q-derivative of the polynomial solution of(32).By using the Bezout’s identity(38)we haveu2(x) n P n(x)ℵn =An(qx)P n(qx)+B n(qx)D q−1P n(qx)(qx) (qx)P n(x)P n(qx)d q x,where n is given in(44).。

Orthogonal Polynomials

Polynomials 7 Some questions leading to classical orthogonal Electrostatics . . . . . . . . . . . . . . . . . . . Polynomial solutions of eigenvalue problems . . Harmonic analysis on spheres and balls . . . . Approximation theory . . . . . . . . . . . . . . 8 Heuristics polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V. Totik
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with special functions, combinatorics and algebra, and it is mainly devoted to concrete orthogonal systems or hierarchies of systems such as the Jacobi, Hahn, Askey-Wilson, . . . polynomials. All the discrete polynomials and the q analogues of classical ones belong to this theory. We will not treat this part; the interested reader can consult the three recent excellent monographs [39] by M. E. H. Ismail, [28] by W. Gautschi and [6] by G. E. Andrews, R. Askey and R. Roy. Much of the present state of the theory of orthogonal polynomials of several variables lies also close to this algebraic part of the theory. To discuss them would take us too far from our main direction; rather we refer the reader to the recent book [24] by C. F. Dunkl and Y. Xu. The other part is the analytical aspect of the theory. Its methods are analytical, and it deals with questions that are typical in analysis, or questions that have emerged in and related to other parts of mathematical analysis. General properties ﬁll a smaller part of the analytic theory, and the greater part falls into two main and extremely rich branches: orthogonal polynomials on the real line and on the circle. The richness is due to some special features of the real line and the circle. Classical real orthogonal polynomials, sometimes in other forms like continued fractions, can be traced back to the 18th century, but their rapid development occurred in the 19th and early 20th century. Orthogonal polynomials on the unit circle are much younger, and their existence is largely due to Szeg˝ o and Geronimus in the ﬁrst half of the 20th century. B. Simon’s recent treatise [80, 81] summarizes and greatly extends what has happened since then. The connection of orthogonal polynomials with other branches of mathematics is truly impressive. Without even trying to be complete, we mention continued fractions, operator theory (Jacobi operators), moment problems, analytic functions (Bieberbach’s conjecture), interpolation, Pad´ e approximation, quadrature, approximation theory, numerical analysis, electrostatics, statistical quantum mechanics, special functions, number theory (irrationality and transcendence), graph theory (matching numbers), combinatorics, random matrices, stochastic processes (birth and death processes; prediction theory), data sorting and compression, Radon transform and computer tomography. This work is a survey on orthogonal polynomials that do not lie on the unit circle. Orthogonal polynomials on the unit circle—both the classical theory and recent contributions—will be hopefully dealt with in a companion article. This work is meant for non-experts, and it therefore contains introductory materials. We have tried to list most of the actively researched ﬁelds not directly connected with orthogonal polynomials on the unit circle, but because of space limitation we have only one or two pages on areas where dozens of papers and several books had been published. As a result, our account is necessarily incomplete. Also, the author’s personal taste and interest is reﬂected in the survey, and the omission of a particular direction or a set of results reﬂects in no way on the importance or quality of the omitted works. For further backgound on orthogonal polynomials, the reader can consult

Higher-order narrow-band array processing

HIGHER-ORDER NARROW-BAND ARRAY PROCESSINGJean-François CardosoTélécom Paris-CNRS URA820-GdR TdSI46rue Barrault,75634Paris Cedex13,France.E-mail:cardoso@sig.enst.frABSTRACTThis communication deals with narrow-band array processing using higher-order cumulants.A versatile tensor formalism is presented and adopted throughout to handle higher-order statistics of complex multivariates.We essentially focus on the use4th-order cumulants for source localization in the spirit of the modern"subspace"techniques. We discuss"full cumulant tensor"solutions and their "downsized"versions in matrix form.Higher-order cumulants are usually advocated for in presence of additive Gaussian noise,but other potential advantages often are overlooked: non-Gaussian sensor noise cancellation,"more sources than sensors"ability,blind identiﬁcation.This is discussed via the notion of fourth-order signal subspace.The paper is not intended as being a review,but other related approaches based on higher-order statistics are brieﬂy reported.1.INTRODUCTIONThe recent renewal of interest for higher-order statistics has been mainly oriented towards time series processing while comparatively less attention was paid to array processing[1]. This paper is devoted to narrow-band array processing.Assume an array of N sensors listening at N s narrow-band sources.The array output is a complex random vector X(t) modelled asX(t)=AS(t)+N(t)t=1,...,N t(1) where N t is the number of available samples,N(t)accounts for additive noise and S(t)is the N s-dimensional vector of source signals.The columns of matrix A are the steering vectors, each one depending on the direction of arrival(DOA)of the wavefront emitted by the corresponding source.We assume in the following that the reader is familiar with this linear model and with the basic concepts associated to it.We will only consider spatial structures:data samples are assumed temporally white.Dealing wih both time and space correlations(multichannel modelling)is out of the scope of this paper.In the past,much effort has been put on processing signals obeying the above linear model and,in recent years,many ideas have been developed for high-resolution estimation based on second-order information.In many circumstances,in telecommunications for instance,signals of interest are signiﬁcantly non-Gaussian:valuable statistical information is present in their higher-order statistics(HOS).An important advantage is obtained with algorithms based on cumulants of order higher than2since these are unaffected by additive Gaussian noise.Of course,this is only true of exact cumulants.Additive noise always is a source of variability in the sample cumulants.We will see in the following that other potential advantages do exist at higher orders,thanks to the larger size of the"statisitics space".The paper is organized according to the following objectives:i)to deﬁne tensor notations for higher-order statistics of complex variables and,hopefully,to demonstrate their versatility.ii)to introduce the notion of fourth-order signal subspace.It is a rather direct generalization of the2nd-order signal subspace but also comes with new properties.We give a preliminary study of its structure.iii)to address the issue of downsizing i.e.the construction of reduced set of higher-order statistics that allows consistent estimation of the signal subspace.iv)to show how these ideas can be used for source localization.v)to report related contributions to the DOA problem using HOS.2.HIGHER-ORDER ALGEBRAAnyone browsing among publications devoted to higher-order statistics would notice that a notation problem deﬁnitely exists.The most popular solution to it consists in using Kronecker products[1]with the advantage of getting equations "looking like"matrix-vector algebra.There are a few drawbacks however.One is that symmetries inherent to moments or cumulants are not easily expressed.For instance, the basic relation(13)can be expressed with Kronecker products only"thanks"to additional multiplication by permutation matrices.Another drawback is that many "stacking"and"unstacking"operations are necessary to arrange higher-order quantities into matrices or vectors.These operations are more about"relabeling"than about "computing".In spite of these drawbacks and of its tendency to hide underlying algebraic structures,if the Kronecker product is almost universally used in higher-order spectral analysis,the reason must be that there is no better solution.We think that this is not so true in theﬁeld of narrow-band array processing because thisﬁeld is more"geometric"than time-series spectral analysis so that indexed tensor notations can be used instead of Kronecker products.Let us say a few words about geometry and then relate it to tensors.Successive samples in time-series analysis,or sensor outputs in narrow-band array processing,are seen as the coordinates of a random vector in some linear space.This vector is then often processed in a very geometric manner. Geometry however is about objects in space:truly geometric ideas are independent of the coordinate system.Because of stationarity a very special basis does exist for time series:the natural basis in which the data vector coordinates just are the signal samples.Time invariances are conveniently expressed in this basis and make its use unavoidable.This is not so true for array processing.The"sensor basis"(where data vectorcoordinates are sensor outputs)certainly is a natural basis,but its use is not really mandatory.As a matter of fact,modern techniques for array processing deﬁnitely have a strong geometricﬂavor with the key concepts of array manifold and of signal subspace and noise subspace,deﬁned in a basis-free manner.When dealing with higher-order statistics,multi-indexed quantities naturally come out.Books on multi-linear algebra are full of these indexﬂocks used to label coordinates(on some basis)of linear applications.Very strikingly,though, strong emphasis is put on the fact that coordinates are just a by-product.They make the writing possible but are used to describe objects which exist independently of their coordinates:geometrical objects are of importance,their coordinates are contingent and depend on an arbitrary basis. Hence,tensor algebra is not very well suited to deal with objects having strong basis-dependent properties(such as stationary times series).As a conclusion,we think that tensor algebra is more adpated to narrow-band array processing than to spectral analysis.It will be used in this paper in its indexed formalism. The reader will notice that there are almost no computations: the indices do much of the work.Of course,it cannot be claimed that indices provide maximal readibility.The best thing to do is probably to switch back and forth between indexed and index-free notations.For instance,to deal with 4th-order statistics of circular complex multivariates,we found easier to deﬁne ad hoc product s[2]acting,like the Kronecker product,in an index-free fashion.When more generality is needed,however,we think that only indexed tensorial notations can offer all the desirable versatility.2.1Tensors on a complex linear spaceLet E be a N-dimensional Hilbert space with an inner product denoted X∗Y and let E∗denote its dual space.We choose an orthonormal basis B for E and denote x i the coordinates of X on this basis.We choose,as a basis for E∗, the dual basis of B which is the basis such that the coordinates of X∗,denoted(this is a convention)as x i,are the complex conjugates of the x i’s.See also[3]about using tensors to deal with HOS of complex multivariates.The"modern"approach to tensor algebra is to deﬁne a (p,q)-tensor as a multilinear form on E∗p×E q.This abstract view allows tensors to be deﬁned independently of any basis. In a more concrete approach,one realizes that,for given p and q,the set of all(p,q)-tensors is a linear space of dimension N p+q on C,so that a(p,q)-tensor,say T,is determined by N p+q complex coordinates.This set of coordinates can be represented by a symbol having p lower indices and q upperindices t i1i2 (i)pj 1j2...jq each index running from1to N.Of course,these coordinates depend on the choice of the basis B and an important concern,when using indexed notations,is to know when a relation between coordinates(hence depending on basis B)actually deﬁnes a relation between tensors(which exist independently of the basis used to express their coordinates).Relations that do not depend on the choice of the basis are termed canonical.In the following,we take the convention(the so called Einstein’s summation convention)that there is an implicit summation over the range of any index repeated once as a subscript and once as a superscript.This convention allows the basic matrix-vector operations to be denoted according to the following equivalences.Vector X→x i,its transpose conjugate X∗→x i,inner product Y∗X→x i y i,outer product XY∗→x i y j,matrix-vector product Y=AX→y i=a i j x j,matrix product C=AB→c i j=a i m b m j,matrix product C=BA →c i j=a m j b i m,left vector-matrix product Y*=X∗A→y i=a j i x j, trace of matrix A→a i i.(Note the irrelevance of symbol positions and index"names"in the tensor expressions).More generally,the summation convention is guaranteed to work "canonically"(a simple example is the trace a i i of a matrix which is known to be a scalar matrix invariant i.e.independent of the basis).These examples,without actual justiﬁcation,showed a (1,1)-tensor involved in various operations.A tensor such as a i j wasﬁrst deﬁned as a bilinear form on E∗×E,that is:given an element of E(a"column vector")x i and an element of E∗(a "row vector")y i,this tensor"works"to yield the scalar a i j y i x j. Without more precaution,we involved it in operations like y i=a i j x j where it is now seen as endomorphism on E.As a matter of fact,tensor theory starts with assessing that such mappings(in our example:from a bilinear form to an endomorphism)actually are canonical isomorphisms.When the number of indices increases,there is an exploding number of such isomorphisms.The beauty of indexed notations is that all possible canonical isomorphisms,that is all the ways in which a tensor can interact with other tensors,are potentially present in a unique symbol.All we have to do is to"let the little indices play".Here are some additional examples.A(2,2)-tensor,say a kl ij,wasﬁrst deﬁned as a multilinear form on E∗×E∗×E×E, i.e.it linearly associates,to any two vectors x i,y i and to any two forms z i,t i,the complex scalar a kl ij x i y j z k t l.This(2-2)-tensor may be used as,loosely speaking,a"linear matrix transform"according to n j i=a jl ik m k l.Now if a jl ik is constructed as a jl ik=u j k v l i,we have n j i=u j k m k l v l i.In standard matrix notations,the relation between matrices M,N,U and V is N=U M V:we get an operator which essentially is the Kronecker product of U and V.2.2Dual tensors,scalar products and tensor normThe dual of,say,a(2,1)-tensor a ij k,is the unique(1,2)-tensor,denoted a−k ij,such that(a ij k x i y j z k)∗=a−k ij x i y j z k for any x i,y i,z k.The dual of a(p,q)-tensor is a(q,p)-tensor denoted by the same symbol with a bar and deﬁned by generalizing the above relation.The notion of"dual of a tensor"generalizes the notion of dual of a vector(hence it would be more consistent to write x−i rather than x i but this is avoided for readibility). Tensor duality allows the tensor norm to be expressed by, going on with the(2,1)-example,|A|2=a ij k a−k ij(2) and the tensor scalar product between tensor a ij k and tensor b ij k is the complex number a−ij k b k ij.If an orthonormal basis is used then a−k ij=˙(a ij k)∗.For(1,1)-tensors,this boils down to the"conjugation-transposition" operation on matrices and the scalar product m j i n−i j is equal to Tr(MN H).Note that we have used a special equal sign"=˙"of our own.This is intended to mean that the identity holds component-wise in a basis speciﬁed elsewhere(here we have speciﬁed"any orthonormal basis").This last relation cannot, anyway,be an identity between tensors since left-hand side and right-hand side do not have the same type.It would also seriously conﬂict with the summation convention.2.3SymmetriesThe issue of symmetry is important,because the symmetries of moments and cumulants translate into tensor symmetries.We found no reference in the litterature about complex tensor symmetry,apart of the(1,1)-case where it reduces to the familiar hermitian plex symmetry has to be speciﬁcally addressed because,while indices can be freely raised,lowered and permuted in the real case,not all index manipulations are allowed in the complex case.The fundamental reason for this is that the canonical mapping from E to E∗,transforming a vector in its dual is not linear but only sesquilinear.This subtle difference has the consequence that only a limited number of index permutations have an intrisic(basis-independent)meaning and correspond to geometric properties.Rather than discussing the issue of tensor symmetry in all its generality,we treat the case of a (2,1)-tensor;extension to(p,q)-tensors is straightforward and is only sketched.Horizontal symmetry.Let a ij k be a(2,1)-tensor.Then there exists a unique(2,1)-tensor b ij k such that a ij k x i y j z k=b ij k y i x j z k for all x i,y i,z k.In the last identity,we have permuted the form arguments,and that made sense because these arguments belong to the same space.Now,we obviously have a ij k=b ji k where indices of the same level are permuted.Note the following:permuting arguments of the same type is always possible allowing us to deﬁne a tensor B starting from a tensor A as above and this deﬁnition is basis-independent.It follows that the related permutation of indices of the same level also is basis-independent.The identity a ij k=b ji k,though expressing a relation between tensor coordinates in some(unspeciﬁed)basis,in fact holds in any basis.More generally,we call horizontal symmetry,any operation transforming a(p,q)-tensor into another(p,q)-tensor by a permutation of upper-indices and/or lower-indices.Horizontal symmetries obviously are automorphisms,and are independent of the basis used to express the permuted coordinates.There are p!q!distinct horizontal symmetries to a(p,q)-tensor.Vertical symmetry.We have seen that taking the dual of a tensor globally exchanges upper and lower indices.Hence (p,p)-tensors and their duals are of the same type,so that the dualization is a symmetry operation on(p,p)-tensors.We talk of vertical symmetry in that case.Super-symmetry.For(p,p)-tensors,by combining vertical and horizontal symmetries,a total of2.p!2is found. This set of symmetries forms a group.We call super-symmetric a tensor invariant under all the elements of this group.The(p,p)-cumulant tensors(see below)are super-symmetric.Note that vertical symmetry is deﬁned through an identity involving dualization of all arguments,while horizontal symmetries involve none.The reader could convince him\herself that expressions of this type could not hold true for all values of arguments if only a part of them was dualized.This is a speciﬁcity of the complex case and explains why there are"only"2.p!2meaningful(i.e.canonical or basis-independent)symmetries and not(2p)!(the number of permutations of2p indices)as in the real case.2.4Tensors as linear applicationsBy selecting p′≤p lower indices and q′≤q upper indices among those of a(p,q)-tensor,it is canonically associated to a linear application.This application maps a(p′,q′)-tensor to a (p−p′,q−q′)-tensor.Here is an example with p=4,q=5, p′=2,q′=1:n jl rstv=a ijkl rstuv m u ik(3) The range of this application is classically deﬁned as the linear space of all(2,4)-tensors that are images through the(4,5)-tensor of some(1,2)-tensor.The rank of this application is deﬁned as the dimension of its range.Clearly,the range of a given tensor is only deﬁned with respect to a speciﬁed subset of its indices.To typographically denote the selected indices, they are indicated with a"×"while the"free"indices,i.e. those pertaining to the image space,are indicated with a"•". For instance,the above application is unambiguously denoted by a×•×••••×•.Outer products and contraction.Without saying,we have already used a few outer products,as,for instance,the Kronecker-like construct a jl ik=u j k v l i.More generally,a tensor outer product consists in constructing a(p1+p2,q1+q2)-tensor by coordinate-wise multiplication between a(p1,q1)-tensor and a(p2,q2)-tensor.Tensor theory guarantees that such an operation actually is basis-independent.The process of contraction consists in setting one upper index and one lower index equal.The resulting implicit summation has the effect of transforming a(p,q)-tensor into a(p-1,q-1)-tensor.This implies that the result actually is an invariant,i.e. basis-independent.Familiar examples of scalar invariants i.e. (0,0)-tensors are the trace a i i and the scalar product a−j i b i j of endomorphisms.When a tensor is identiﬁed to a linear application,as in(3),the result might be seen as p′+q′contractions of the outer product of the involved tensors.Eigen-structure of(2,2)-tensors.We specialize to the case of(2,2)-tensors,which is of special interest in the following.We extend the classical result that if a(1,1)-tensor is self-adjoint,i.e.if a−i j=a i j,then it admits N real eigen-values, N orthonormal eigen-vectors,and the eigen-structurea i j=v=1ΣNλv v i v j(4)whereλv is the real v-th eigen-value and where v i denotes the i-th coordinate of the v-th eigen-vector.Let us extend this result to(2,2)-tensors.We select the last lower and upper indices of a(2,2)-tensor g jl ik to use it as an endomorphism on (1,1)-tensors according to the pattern g•×•×:n j i=g jl ik m k l(5) According to this index pattern,the adjoint appplication(see appendix)is h jl ik=g−lj ki.Any tensor verifying g jl ik=g−lj ki is self-adjoint for that pattern and,consequently admits N2real eigen-values and N2orthonormal eigen-matrices.It can be expressed by the following eigen-decomposition:g jl ik=e=1ΣN2µe e j i e−l k(6)whereµe denotes the e-th eigen-value and e i j denotes the e-th eigen-matrix.If,in addition,the tensor satisﬁes the following horizontal symmetry:g jl ik=g lj ki,then the eigen-matrices can be chosen hermitian:e l k=e−l k[4].2.5Delta symbolsWe denote byδi1 (i)pj 1...jq a symbol which is equal to1if allindices are equal and zero otherwise.If in some orthonormal basis,a(1-1)-tensor has coordinates equal toδi j,then it is the identity(1-1)-tensor and its coordinates are equal toδi j in any other basis.It follows thatδi j canonically determines a(1,1)-tensor.If in some basis,a(2,2)-tensor,for instance,has coordinates equal toδij kl,then it has different coordinates in (allmost all)other basis.It follows thatδij kl does not deﬁne a (2,2)-tensor unless a particular basis is speciﬁed.3.HIGHER-ORDER STATISTICSIn this section,we introduce tensor notations for higher-order moments and cumulants of complex multivariates,derive cumulant expressions for the linear data model(1)and investigate the resulting cumulant tensor structures.3.1Higher-Order Statistics for complex multivariatesWe essentialy adapt to the complex case the formalism given in[5]for real multivariates.We consider a complex random vector X with coordinates x i.Moments.The(p,q)-moments of X form a(p,q)-tensor deﬁned asµi1i2 (i)pj 1j2...jq=∆E{x i1x i2...xipx j1x j2...x j q}(7)We insist that this deﬁnition actually determines a(p,q)-tensor since it is the expected value of an outer product of(1,0)-and (0-1)-tensors.Moments of any orders clearly are horizontally symmetric tensors and,in addition,(p,p)-moments are vertically symmetric,hence super-symmetric.Of particular relevance will be in the following2nd-and4th-order moments:µi j=E{x i x j}(8)µi l j k=E{x i x j x l x k}(9) Circularity.The complex circularity property is conveniently deﬁned by:"the(p,q)-moments are null except if p=q".The"circular"denomination comes from the following fact.If the complex multivariate is multiplied by a complex phase term exp iφ,then the(p,q)-moments are multiplied by exp i(p−q)φ.Since the phase term affects only the moments with p≠q,it follows that a circular multivariate has all moments invariant under a phase rotation.Cumulants.Cross-cumulants of a set of random variables can be deﬁned as symmetrized products of moments of same and lower orders.In tensor form,these products take the form of outer products.It follows that(p,q)-cumulants deﬁned byκi1 (i)pj 1...jq=∆Cum(x i1,...,x ip,x j1,...,x j q)(10)also are(p,q)-tensors.Since cross-cumulants are symmetric in their arguments,cumulant tensors also are symmetric.More precisely,cumulants of any(p,q)-order are horizontally symmetric tensors and(p,p)-cumulant tensors are,in addition,super-symmetric.The(1,1)-cumulant tensor is the familiar covariance of a complex multivariate.We will call quadricovariance the4th-order(2,2)-cumulant tensor.We will often use it as an application by selecting one upper index and one lower index. It is then seen as a linear applicationκ•×•×on(1,1)-tensors.The quadricovariance being super-symmetric,it is not necessary to specify which upper index and lower index are selected.For zero-mean variables,the classical relations between 2nd-and4th-order moments and cumulants readκi j=µi j(11)κi l j k=µi l j k−µi jµl k−µi kµl j−µi lµj k(12) and for circular variables,the latter further reduces to κi l j k=µi l j k−µi jµl k−µi kµl j.(13)3.2Signal cumulant structureWe consider the general form of cumulants for the signal part of the data,that is for the noiseless part of eq.(1)which is rewritten in indexed notations,together with its dual,asy i=a i p s p(14)y i=a−p i s p(15)where we have dropped explicit dependence on t.With these two equations,we have made an important extension to the prior deﬁnition of tensors since we now have indices related to different spaces.Indices labeled with i in(14,15)refer to the N-dimensional"array output space"while those labelled with p refer to the N s-dimensional"source space".This extension does not raise any difﬁculty.Of course,only indices referring to the same space can be contracted.Letκand S denote the cumulants of signals and sources respectively.It sufﬁces to invoke the linearity of cumulants to getκi1...ipj1...jq=S m1...mpn1...nq a i1m1...a ipmp a−n1j1...a−nqjq(16) As a matter of fact this is just the general transformation law of (p,q)-tensors(hence of moments or cumulants)under a linear transformation.These relations hold regardless of circularity. In a more readable form,they are,at orders(1,1)and(2,2):κj i=a jβSβαa−αi(17)κjl ik=a−αi a jβSβδαγa−γk a lδ(18)In the following,we will focus on the(2,2)-cumulant tensor (the quadricovariance)because,among the higher-order cumulant tensors which do not vanish in the circular case,it is the one with the lowest order.Hence the last equation will be our"raw material".4.FOURTH-ORDER SIGNAL SUBSPACEThe concept of signal subspace has proved very helpful in the processing of the covariance.In this section,we introduce its4th-order conterpart.It is quite a natural extension but, interestingly enough,it is much more"informative"than the 2nd-order signal subspace since it is more sensitive to the nature of source correlations and to the form of the steering vectors.Before going to4th-order,weﬁrst recall the classical results on the(1,1)-cumulant tensor(the covariance)to show analogies and differences between2nd-order and4th-order cases.The covariance structure(17)is the index form of the familiar matrix relation K=ASA H.The covariance range is called the signal subspace.It is well known that for full rank source covariance,the signal subspace is the range of a•×,i.e. the span of the steering vectors.If steering vectors are linearly independent,the dimension of the signal subspace is then equal to the number of sources.Note that the signal subspace is identical in the case of independent sources and in the case of correlated sources(as long as they are not fully correlated).Let us now turn to the4th-order case.We call4th-order signal subspace(FOSS)the range of the quadricovariance κ•×•×.It is the purpose of this section to give aﬁrst account of the structure of the FOSS.4.1Correlated sourcesThe similarity between(17)and(18)shows that,if the source quadricovariance S•×•×is full rank,then the FOSS is the range of a•×a−×•.Hence the FOSS appears as the matrix space spanned by the outer products of all possible pairs of steering vectors[6].In other words,the relation between2nd-order and 4th-order signal subspace is:"the FOSS is the linear space of all operators equal to their restriction on the signal subspace". If the N s steering vectors are linearly independent,the FOSS has dimension N s2.Note in that case that the FOSS cannot be "any"matrix space with dimension N s2since it has the distinctive feature of being invariant under the composition product.Without any further assumptions,it is not possible to be more speciﬁc because the actual FOSS structure ultimately depends on source quadricovariance.So it is worth considering some particular source correlation models.4.2Independent sourcesWeﬁrst consider the case of independent sources.The source cumulant tensor then has null coordinates(in its natural basis)unless all indices are equal.This highly degenerate form calls for special notation because the many summations on source indices in the above equation then reduces to a simple summation through each source.Let us denote by s i the i-th coordinate of the s-th steering vector(s i=˙a i s)and byσs and k s respectively the scalar(1,1)-and(2,2)-cumulant of source s(respectively"power"and"kurtosis").The summation on sources has now to be explicit:κj i=sΣσs s i s j(19)κjl ik=sΣk s s i s j s k s l(20)Note that bothσs and k s are real numbers and that the k s can be positive or negative.Let us call steering matrix the outer product s j s i,of a steering vector s i.The above expression shows that the application on any matrix ofκ•×•×yields a linear combination of steering matrices.Hence the FOSS is the span of the steering matrices.If the steering matrices are linearly independent,the FOSS has then exactly dimension N s for statistically independent sources with non-zero kurtosis.This is theﬁrst important difference between the2nd-and4th-order cases:partial correlation between sources leaves the2nd-order signal subspace unchanged,while the FOSS dimension goes from N s to possibly N s2.Also note that i)linearly independent steering vectors gives linearly independent steering matrices ii)the converse is false:more sources than sensors yield linearly dependent steering vectors but the corresponding steering matrices are not necessarily linearly dependent.When sources are independent,the FOSS has a fascinating feature.It is linearly generated by N s steering matrices which also are,by deﬁnition,rank-one matrices.It can be demonstrated[2]that under mild conditions,these are the only (up to a scalar factor)rank-one matrices in the FOSS.It follows that steering matrices(hence steering vectors)can be retrieved from the FOSS without even knowing the array manifold(blind identiﬁcation)by looking for the rank-one matrices of the FOSS.4.3Special correlation casesTo gain further insight into signal quadricovariance and its range,we now work out a few scenarios of correlation.For sake of simplicity we consider only two sources emitting signals s1(t)and s2(t)with steering vectors denoted by p i and q i respectively.The general case of4th-order correlation is when the source quadricovariance is full rank.The FOSS has then22=4dimensions and is spanned by all the outer products p i p j,q i q j,p i q j and q i p j.If signals s1and s2are independent, then the FOSS has only two dimensions and is generated by p i p j and q i q j.Linearly correlated sources.If two independent signals are linearly mixed by a regular2×2matrix,we get two correlated signals.If s1and s2are such a linear mix of independent signals,the signal quadricovariance"knows" about it because the array output still is the sum of two independent signals,each one modulating mixed steering vectors.Signal quadricovariance then keeps a rank equal to2 and its range is spanned by the outer products of the correpondingly mixed steering vectors.Amplitude correlated Gaussian signals Let s1be a circular Gaussian signal with unit power.Let s2have the same amplitude as s1but an independent phase,uniformly distributed on[−π,+π].Both s1and s2are Gaussian variables but not jointly Gaussian.Quadricovariance is easily computed to beκjl ik=p i p j q k q l+p i q j q k p l+q i p j p k q l+q i q j p k p l and clearly has rank4(consider its span).Real signals.Let us conclude with an example of intermediate rank which is given by an"anti-circular"case, where s1and s2are assumed to have constant phases.These phase terms can be transfered to the steering vectors so that the source signals appear to be real.It is then easily shown that the FOSS is3-dimensional and is spanned only by p i p j,q i q j and p i q j+q i p j.More generally for N s real sources,cumulant symmetries in the real case force the source quadricovariance to have a range with at most N s(N s+1)/2dimensions.The simple reason is that symmetric real matrices form a linear matrix subspace on R while complex hermitian matrices do not form a linear subspace on C.Phase coupled harmonics show another type of source correlation.Swindelhurst and Kailath[7]have considered in that case the third-order moment matrix which,for zero-mean random processes,is identical to the cumulant.They have investigated how the ranks and ranges ofκ••×andκ××•are。

On the role of meridional flows in flux transport dynamo models

Astronomy & Astrophysics manuscript no. 7070 February 2, 2008
c ESO 2008
On the role of meridional ﬂows in ﬂux transport dyun Allan Sacha
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Cattaneo & Hughes 2001; Ossendrijver 2003). Developing numerical models of the solar dynamo has thus been a very active ﬁeld of research. This has mainly involved two types of numerical experiments: – kinematic solar dynamo models that solve only the induction equation in its mean ﬁeld approximation and assume the velocity ﬁeld as given (Steenbeck & Krause 1969; Roberts 1972; Stix 1976; Moﬀat 1978; Krause & Radler 1980; see Charbonneau 2005 and Solanki et al. 2006 for recent reviews). These models rely on the parametrization of two important eﬀects that are thought to be at the origin of the solar global dynamo, the α and Ω eﬀects. They provide a useful and fast tool to model the solar 22-yr magnetic cycle and its associated butterﬂy diagram since no feedback from the Laplace force on the motion is accounted for. – or dynamical solar dynamo models that solve explicitly the full set of MHD equations (Gilman 1983; Glatzmaier 1985; Cattaneo 1999; Brun et al. 2004). These models selfconsistently compute all the physical processes in three dimensions allowing signiﬁcant progress to be made on the intricate interactions operating in a turbulent magnetized plasma. The cost of 3D models and the large number of degrees of freedom needed to model the whole Sun make it diﬃcult, as of today, to provide quantitative predictions such as the cycle period. Clearly, both approaches are complementary and are needed to better understand the magnetic solar activity. Since the original

On quasicrystal Lie algebras

On quasicrystal Lie algebrasVolodymyr Mazorchuk2000Mathematics Subject Classiﬁcation:17B68,17B10,17B81Key words:aperiodic Virasoro algebra,highest weight module,Shapovalov form,Kac determinantAbstractWe realize the aperiodic Witt and Virasoro algebras as well as other quasicrystal Lie algebras as factoralgebras of some subalgebras of the higher rank Virasoro alge-bras.This realization allows us to generalize the notion of quasicrystal Lie algebras.In the case when the constructed algebra admits a conjugation,we compute the Kacdeterminant for the Shapovalov form on the corresponding Verma modules.In thecase of the aperiodic Virasoro algebra this proves the conjecture of R.Twarock.1IntroductionThis paper has grown up from my attempt to understand the recently introduced notion of quasicrystal Lie algebras and the aperiodic Witt and Virasoro algebras,[PPT,T1]. These algebras form a new family of inﬁnite-dimensional Lie algebras,whose generators are indexed by points of an aperiodic set(which is in fact a one-dimensional cut-and-project quasicrystal,an object,intensively studied by many authors,see e.g.[Ka,K,R] and references therein).Quasicrystal Lie algebras and their representations were studied in[PT,PPT,T1]and in[T2,T3]some applications of these algebras to construction of some integrable models in quantum mechanics were given.However,there are many important questions about the quasicrystal Lie algebras,which are still open.For example,in[T1]the author constructs a triangular decomposition for the aperiodic Virasoro algebra,hence constructing Verma modules,and conjectures a formula for the Kac determinant of the Shapovalov form on these modules.This formula in important both for description of simple highest weight modules and for picking up those of them which can be unitarizable,which is the question of primary interest in physical applications.It was clear from the veryﬁrst deﬁnition of quasicrystal Lie algebras,that this notion should be closely connected with the notion of the higher rank Virasoro algebras,deﬁned in [PZ].The major diﬀerence between these algebras is that the indexing set for quasicrystal Lie algebras is a discreet subset of R while for the higher rank Virasoro algebras the corresponding set is everywhere dense.In the present paper we establish this connection1by realizing quasicrystal Lie algebras as factoralgebras of some subalgebras of the higher rank Virasoro algebras.This realization allows us to generalize quasicrystal algebras in several directions,preserving the property to have a discreet indexing set.Moreover,the notion and construction of triangular decomposition for these algebras appears naturally in this framework.Further,we discuss the existence of conjugation on constructed algebras, which pairs the components of the positive and negative part.In the case,when such pairing exists,the deﬁnition of the Shapovalov form on Verma modules(see[S,MP])is straightforward and we compute the Kac determinant(see[S,KK,MP,KR])of this form. In the case of the aperiodic Virasoro algebras this proves[T1,Conjecture V.7].The paper is organized as follows:in Section2and Section3we remind the deﬁnitions of quasicrystal Lie algebras and higher rank Virasoro algebras.We give a realization of quasicrystal Lie algebras as factoralgebras of certain subalgebras of the higher rank Virasoro algebras in Section4and use it to construct parabolic and triangular decompositions of our algebras in Section5.In Section6we study the Verma modules and,in particular, calculate the determinant of the Shapovalov form on them.Weﬁnish with discussing several generalizations of our construction in Section7and Section8.2Quasicrystal Lie algebras√Denote by(·)#the unique non-trivial automorphism of theﬁeld Q(5.LetΩbe a non-empty,connected and bounded real set, whose set of inner points does not contain0.Setτ=15).Then the quasicrystal Σ(Ω),associated withΩ,is the set of all x∈Z[τ],such that x#∈Ω.The quasicrystal Lie algebra L(Ω),associated withΩ,is deﬁned as follows(see[PPT]):it is generated over F by L x,x∈Σ(Ω),with the Lie bracket deﬁned via[L x,L y]= (y−x)L x+y,x+y∈Σ(Ω)0,otherwise.To deﬁne the aperiodic Witt and Virasoro algebras as it is done in[T1],we introduce the mapϕ:Z[τ]→Z,which sends x=a+bτtoϕ(x)=b.Then the aperiodic Witt algebra AW([0,1],F)is generated over F by L x,x∈Σ([0,1]),with the Lie bracket deﬁned via[L x,L y]= (ϕ(y)−ϕ(x))L x+y,x+y∈Σ([0,1])0,otherwise.By[T1,Theorem III.4],the algebra AW([0,1],F)admits the unique central extension AV([0,1],F),called the aperiodic Virasoro algebra,which is generated over F by L x,x∈Σ([0,1]),and c,with the Lie bracket deﬁned via[L x,L y]= (ϕ(y)−ϕ(x))L x+y+δϕ(x),−ϕ(y)ϕ(x)3−ϕ(x)33The higher rank Virasoro algebrasLet P denote the free abelian group Z k ofﬁnite rank k andψ:P→(F,+)be a group monomorphism.The rank k Virasoro algebra V(ψ,F),associated withψ,is generated over F by elements e x,x∈P,and central c,with the Lie bracket deﬁned viaψ(x)3−ψ(x)3[e x,e y]=(ψ(y)−ψ(x))e x+y+δx,−yadmissible order on P.Indeed,<is obviously antisymmetric,antireﬂexiv and transitive.So,it is a partial order.But from the deﬁnition it also follows immediately,that<is linear. Further,for any a<b in P and c∈P we have(ψ(a+c)−ψ(b+c))#=(ψ(a+c−b−c))#=(ψ(a−b))#=(ψ(a)−ψ(b))#<0and hence a+c<b+c,thus<is compatible withthe addition in P.Finally,if0<a<b then0<(ψ(a))#and hence there always exists k∈N such that(ψ(b)−ψ(ka))#=(ψ(b))#−k(ψ(a))#<0,which shows that the order is admissible.Consider the rank2Witt algebra G=W(ψ,F).Without loss of generality we can assume thatΩ⊂R0,as in other case we can work with the order,opposite to<.AsΩis a connected bounded subset of R,it has one of the following four forms:[a,b],(a,b],[a,b), (a,b)for some non-negative real a,b.We deﬁne I and J as follows:I is generated by all e x such thatψ(x)#>a(resp.ψ(x)# a)if a∈Ω(resp.a∈Ω);and J is generated by all e x such thatψ(x)#>b(resp.ψ(x)# b)if b∈Ω(resp.b∈Ω).From the deﬁnition it follows immediately that both I and J are non-negative ideals of P with respect to <.Hence,the algebras L(G,<,I)and L(G,<,J)are well-deﬁned subalgebras of G and L(G,<,J)⊂L(G,<,I)by deﬁnition.Now we show that L(G,<,J)is actually a ideal of L(G,<,I).Indeed,if x∈I andy∈J we get that x+y∈J as J is an ideal of P and x∈P0+.Hence[e x,e y]∈L(G,<,J) for any e x∈L(G,<,I)and e y∈L(G,<,J).Finally,we consider the map f:L(G,<,I)→L(Ω)deﬁned byf(e x)= L x,x ∈Ω0,otherwise.From the deﬁnition of the Lie brackets in L(Ω)(Section2)and in G(Section3)weimmediately get that f is a Lie algebra homomorphisms.Moreover,it is also clear that its kernel coincides with L(G,<,J).This completes the proof.Theorem1motivates the following deﬁnition:let G=W(P,ψ)be a higher rank Witt algebra(it is important here that k>1,i.e.that G is not the classical Witt algebra), <be an admissible order on P,and I⊃J be two non-negative ideals of P with respect to the order<.Then we deﬁne the Lie algebra A(P,ψ,<,I,J)of quasicrystal type as the quotient algebra L(G,<,I)/L(G,<,J).In particular,all quasicrystal Lie algebras are Lie algebra of quasicrystal type.Now we can formulate some basic properties of Lie algebras of quasicrystal type and we see that these algebras share a lot of properties of classical quasicrystal Lie algebras.We start with the following easy observation.Lemma 1.Let<be an admissible order on Z k.Then there exists a homomorphism,σ:P→R,such thatσ(P±)⊂R±.Proof.Identify P with Z k⊂R ing the description of admissible orders on an abelian group from[Z],weﬁnd a hyperplane,H,of R k,such that P+coincides with the set of points from Z k,which are settled on the same side with respect to H.Thenσcan be taken,e.g.the projection on H⊥with respect to H(here R k is considered as an Euclidean space in a natural way).4For given G=W(P,ψ)and<weﬁx someσ,existing by Lemma1.We deﬁne a= inf x∈I(σ(x))and b=inf x∈J(σ(x))and will use this notation in the following statement. Proposition1.Let A(P,ψ,<,I,J)be a Lie algebra of quasicrystal type.1.A(P,ψ,<,I,J)is abelian if and only if2a b.2.A(P,ψ,<,I,J)has non-trivial center if and only if a=0.3.A(P,ψ,<,I,J)is nilpotent if and only if a>0and J=∅.4.The algebra A(P,ψ,<,I,J)is perfect if an only if a=0and0∈I.5.If J=∅then anyﬁnite set of elements in A(P,ψ,<,I,J)generates aﬁnite-dimensional Lie subalgebra of A(P,ψ,<,I,J).In particular,A(P,ψ,<,I,J)has ﬁnite-dimensional subalgebras of arbitrary non-negative dimension.Proof.All statements are easy corollaries from the additivity of indices of generating ele-ments under the Lie bracket.Indeed,with this remark theﬁrst statement reduces to the fact that x a and y a implies x+y 2a b;the second one reduces to the fact that for any x a and y>b−a holds x+y>b;and the third one reduces to the fact that for ka>b we have kx>b for any x a.If a=0,the algebra A(P,ψ,<,I,J)is nilpotent by statement three and hence not perfect.It is also clear that it is impossible to get0as a result of the Lie operation.But if a=0and0∈I,then L(G,<,I)=G+,σ(P+)is dense in R+and hence for any x>0there are y,z∈P+such that y+z=x.This implies that A(P,ψ,<,I,J)is perfect in this case and hence the property four.Theﬁrst part of the last statement is equivalent to the trivial statement that aﬁnite subset of R+generates an additive semigroup,whose intersection with any bounded set is ﬁnite.To prove the second part it is suﬃcient to consider the span of e ix,i=1,...,n, such that nx<b and(n+1)x>b.This completes the proof.For example,to realize the aperiodic Witt algebra AW([0,1]),deﬁned in[T1],as a Lie algebra of quasicrystal type,one should take P=Z2,ψbeing the projection on the second coordinate;<deﬁned by x<y if and only if the inner product of y−x with(1,15)) is greater than zero;I=P0+;J={x∈P:(1,0)<x}.5Standard and non-standard triangular decomposi-tionsThe realization of Lie algebras of quasicrystal type,obtained in the previous section al-lows us to adopt the technique from[M]to construct various triangular and parabolic decompositions of these algebras.The general procedure will look as follows.Let A=A(P,ψ,<,I,J)be a Lie algebra of quasicrystal type.Abusing notation we will denote by e x,x∈I\J,the generators of A.Choose any linear pre-order, ,on the abelian5group P ,which is diﬀerent from <and its opposite.Deﬁne A ±as the Lie subalgebras of A ,generated by all e x ,0≺±x ,and set A 0to be the Lie subalgebra of A ,generated by all e x ,0 x and x 0.We get the following obvious fact.Lemma 2.A =A −⊕A 0⊕A +.Proof.Clearly,A =A −+A 0+A +.The fact that this is actually a direct sum decomposition follows easily from the the property x y implies x +z y +z .It is natural to call the decomposition A =A −⊕A 0⊕A +parabolic decomposition of A ,associated with .Given a parabolic decomposition and a simple A 0-module,V ,one can extend V to an A 0⊕A +-module with the trivial action of A +and construct the associated generalized Verma module M (V )as follows:M (V )=U (A )⊗U (0⊕+)V .If A 0happens to be rather special,it is natural to rename the corresponding parabolic decomposition into triangular decomposition .However this is a subtle question and the hierarchy I give here represent only my point of view and is inspired by the corresponding notions for the higher rank Virasoro algebra ([M]).We will say that the decomposition A =A −⊕A 0⊕A +is a standard triangular de-composition provided A 0=F e x for some x ∈P ,which is not maximal in I \J .We call A =A −⊕A 0⊕A +the non-standard triangular decomposition provided A 0is a commutative Lie algebra and the parabolic decomposition fails to be a standard triangular.In the case of triangular decomposition generalized Verma modules become classical Verma modules as in this case dim(V )=1.The ﬁrst case is natural and corresponds to triangular decompositions of the higher rank Virasoro algebras,[M].Actually,here one has to be careful because,depending on whether ≺satisﬁes the Archimed law or not,one can further distinguish two cases of standard triangular decomposition.We will not do this,as we will not study the diﬀerence between the corresponding situations.But the second case has a striking diﬀerence from the ﬁrst one and comes from the deﬁnition of triangular decomposition for the aperiodic Virasoro algebra in [T1].This means that the triangular decomposition for the aperiodic Virasoro algebra,constructed in [T1]is an example of a non-standard triangular decomposition.We now will study analogous situations in more detail.We retain the notation for σ,a,b from the previous section and further assume that a =0and that that there is an element,e u ∈A ,such that σ(u )=b and ψ(u )=0.Since <is an admissible order,such element is unique and we retain the notation e u for it.Lemma 3.Under the above assumptions we consider the vectorspace A =A ⊕F c .Then the formula [e x +a c ,e y +b c ]=[e x ,e y ]+ψ(x )3−ψ(x )The algebra A,constructed in Lemma3is a natural generalization of the aperiodic Virasoro algebra from[T1].In particular,the aperiodic Virasoro algebra coincides with A for A constructed in the end of the previous section.However,if e.g.the rank of P is bigger than two,we get an example of A ,which diﬀers from the aperiodic Virasoro algebra.We will call algebras A the Virasoro-like algebras of quasicrystal type.Assigning the element c index u we easily transfer the notions of parabolic and both standard and non-standard triangular decompositions on algebra A .If P has rank two, then,up to taking the opposite order,the non-standard triangular decomposition of A, such that A 0contains e u,is unique,and in the case of the aperiodic Virasoro algebra this coincides with the triangular decomposition,constructed in[T1,Section V].For the rank two case once can easily construct example of A such that with respect to the unique natural non-standard triangular decomposition,mentioned above,dim(A 0)is an arbitrary positive integer.Hence even in rank two case one gets a lot of examples of A ,diﬀerent from the aperiodic Virasoro algebras.All these algebras will have discrete aperiodic root systems,and,if considered as graded by the action of e0,all roots will be multiple with multiplicity dim(A 0)−2.In the case of the aperiodic Virasoro algebra we have dim(A 0)=3 and hence all roots(with non-zero action of e0)are multiplicity free.We will discuss this situation in more details in the next section,when we will deﬁne the Shapovalov Form on the Verma modules and compute its determinant.6Shapovalov form and Kac determinantIn this section we present several results on the structure of Verma modules over Lie and Virasoro-like algebras of quasicrystal type.As in the case of the Witt and the Virasoro algebras,the representation theory the last one is more complicated,which,in particular, gives a bigger variety of simple highest weight modules.Our main tool in the case of the Virasoro-like algebras of quasicrystal type and the corresponding Lie algebras of quasicrys-tal type will be the Shapovalov form on Verma modules,ﬁrst deﬁned in[S]for simple ﬁnite-dimensional Lie algebras.However,we start with more elementary general case of Lie algebras of quasicrystal type,which happens to be really trivial.Before starting we just note that in this section we always assume that F is an algebraically closedﬁeld of characteristic zero.We recall that,given a triangular decomposition,A=A−⊕A0⊕A+,an A-module,M, is called a highest weight module,if there exists a generator,v∈M,such that A+v=0. Proposition2.All simple highest weight modules over a Lie algebras of quasicrystal type, which correspond to a standard triangular decompositions with A0=F e x⊂[A,A]are one-dimensional.In particular,corresponding Verma modules are always reducible. Proof.This is a direct corollary of A0=F e x and e x∈[A,A].Proposition3.All Verma modules over a Lie algebras of quasicrystal type,which corre-spond to a standard triangular decompositions with A0=F e x⊂[A,A],are reducible.The7corresponding unique simple quotients are one-dimensional if and only if the eigenvalue ofe x on the primitive generator of the module is zero.Otherwise they are inﬁnite-dimensional. Proof.Let v be the canonical generator of the Verma module in question.The reducibilityfollows from the fact that x is not maximal in I\J,and hence there are inﬁnitely manyelements y∈P−satisfying e x∈[e y,A],which implies that U(A)e y v is a proper submodule of the Verma module.The second statement follows considering the set of elements e y,y∈P−,satisfying e x∈[e y,A],which is obviously inﬁnite.So,we can now move on to the case of non-standard triangular decomposition.Firstwe reduce our consideration to the natural case of weight modules withﬁnite-dimensionalweight spaces,which corresponds to the situation,when the root system of A is discrete. This is only possible in the case when P Z2.Here our main tool will be the Shapovalov form and to be able to work with it we will also need the following assumptions from the previous section:e0∈A;and there is e u∈A,such thatσ(u)=b andψ(u)=0.As it was mentioned above,this situation covers,for example,the case of the aperiodic Witt algebra.Since in the case of the algebra A the arguments will be absolutely the same,we consider both cases simultaneously with all the notation for the algebra A .The case of A is then easily obtained by factoring c=0out.We deﬁne the conjugation on P viaω(x)=u−x and it follows immediately from our assumptions that e x∈A implies eω(x)∈A .However,is easily to see thatωdoes not extend to an(anti)involution on A .We note thatσ(ω(x))=b−σ(x).We recall that the algebra A is graded by the adjoint action of e0(or,more general,A 0)and for C α=0the dimension of A αis either0or dim(A 0)−2(dim(A0)−1inthe case of algebra A).We denote by∆the set of all(non-zero)roots of A with respectto this action and by∆±the sets of all positive and negative roots corresponding to our triangular decomposition.Obviously,ωextends to a linear bijection A α→A −αfor any α∈∆∪{0}.As A 0is commutative,simple A 0-modules are one-dimensional and have the form Vλ,λ∈(A 0)∗,where the action is deﬁned via g(v)=λ(g)v for v∈Vλand g∈A 0.Let vλdenote a canonical generator of M(Vλ).Let∆ (resp.∆ ±)denote the semigroup,generated by∆(resp.∆±).Then the module M(Vλ)is a weight module with respect to A 0with the supportλ∪λ−∆ +.All weight spaces of M(Vλ)areﬁnite dimensional.Moreover,M(Vλ)is isomorphic to U(A −)vλas a vectorspace.The∆±-gradation of A ±extends to the∆ ±-gradation of U(A ±)and,in the aniinvolutive way,ωextends to a linear componentwise isomorphism from U(A +)to U(A −)and back, which matches U(A +)αwith U(A −)−α.Forµ∈Supp(M(Vλ)),µ=λ−ν,ν∈∆ +,we deﬁne the Shapovalov form Fλ,νon M(Vλ)µby setting that Fλ,ν(fvλ,gvλ),f,g∈U(A −)−ν,equals the coeﬃcient ofω(f)gvλ∈M(Vλ)λ,written in the basis{vλ}.The following properties of Fλ=⊕ν∈∆Fλ,νare standard and the reader can consult[KK,MP]for the arguments.8Lemma4. 1.M(Vλ)is simple if and only if Fλis non-degenerate.2.The kernel of Fλcoincides with the unique maximal submodule of M(Vλ).Hence in order to study the reducibility of M(Vλ)it is suﬃcient to compute the determinant of Fλ,νfor allλandν.To be able to do this we consider the followingmonomial generators of U(A −)−ν:G=G(ν)={g(x1,...,x k)=e x1...e xk:x i∈∆−;i x i=−ν;σ(x i) σ(x i+1)}.We deﬁne the linear order on this set of generators as the lexicographical order with respect to the values ofσ(x i).The key property of this construction is the following.Lemma 5.If g(x1,...,x k)∈G and g(y1,...,y m)∈G are such that g(x1,...,x k) g(y1,...,y m).Then Fλ,ν(g(x1,...,x k)vλ,g(y1,...,y m)vλ)=0.Proof.Let i be minimal such thatσ(x i)<σ(y i).Thenσ(ω(x i))>b−σ(y i)and henceeω(xi)commutes with e yiand thus with all e yj,j i,since for such j we haveσ(y j) σ(y i)form the deﬁnition of G.For j<i we have x j=u j and thus[eω(xj),e yj]∈A 0.We canwriteeω(xi)(ω(e x1...e xi−1)e y1...e yi−1)e yi...e ymvλ==ε(ω(e x1...e xi−1)e y1...e yi−1)eω(xi)e yi...e ymvλ+other termsfor someε∈F,where in other terms some eω(xj),j<i,occurs already after the corre-sponding e yj .As eω(xi)commutes with all e yj,j i,we get that theﬁrst summand equalszero.Now consider one of the other terms and let eω(xj)be the factor occurring most to theright in the monomial.This means,in particular,that for s<j this monomial contains[eω(xs),e ys],which are the elements of A 0and thus,up to a scalar factor,can be movedto the left.In particular,σ(ω(x s))is the biggest value among all others occurring in thismonomial.If the element e y,standing next to eω(xj)satisﬁes y=x j,this means that eω(xj)commutes with e y and hence the monomial contributes0to the global sum.Otherwise thenumber of factors,standing to the right from eω(xj),which equal x j,is less than the samenumber before the last commutation.Hence induction in this number reduces the problem to the case y=x j thus proving that all monomials occurring in other terms contribute0 to the global sum.From this it follows directly that Fλ,ν(g(x1,...,x k)vλ,g(y1,...,y m)vλ)=0,which com-pletes the proof.From Lemma5we immediately get the following statement,which,in particular,proves [T1,Conjecture V.7].Corollary 1.The determinant of Fλ,νcoincides with the product of diagonal elements Fλ,ν(g(x1,...,x k)vλ,g(x1,...,x k)vλ).9Now we can formulate the computation results for the determinant of the Shapovalov form and the corresponding corollaries for the structure of M(Vλ).Denote by P the set of all non-zero x∈P such that e x∈A is non-zero.Then the decomposition∆=∆−∪∆+ induces a decomposition P =P −∪P +.Forν∈∆ +and x∈P −we denote by pν(x)thenumber of occurrences of e x as factors in the canonical decomposition of all monomials in G(ν).Theorem2.Up to a non-zero constant the determinant of Fλ,νequalsx∈P − λ(e u)−ψ(x)2−112c,asψ(u)=0.Moreover,[e u−x,e u]is in fact central in A .Hence,we can movethe non-zero factor2ψ(x)out and get that,up to a non-zero constant factor,we haveFλ,µ(g(x1,...,x k)vλ,g(x1,...,x k)vλ)=ki=1 λ(e u)−ψ(x i)2−1Proof.Under these conditions all factors of the diagonal elements of the matrix of the Shapovalov form are non-negative and hence all leading minors are non-negative as well. This implies the statement.Using these results we also get some information about highest weight modules,asso-ciated with standard triangular decompositions.Corollary4.The dimensions of the weight spaces of inﬁnite-dimensional highest weight modules over Lie algebras of quasicrystal type,associated with standard triangular decom-positions,are not uniformly bounded.Proof.Let A0=F e x be the zero component of the given standard triangular decomposition. Then we can factor our an ideal of A such that the factoralgebra is still of quasicrystal type,but the element x became maximal in the corresponding I\J,and hence the induced triangular decomposition became non-standard.Now we haveλ(e x)=0and hence the corresponding Verma module over this algebra is simple and the dimensions of its weight spaces are obviously unbounded.Buy this module naturally embeds(as a vector subspace) into the simple highest weight module,which we started with.7Further generalizations of quasicrystal Lie algebras Geometrical realization of the algebra A,obtained in Section4,motivates the following generalization of the class of Lie algebras of quasicrystal type.We consider arbitrary rank n Witt algebra G=G(P,ψ)with P Z n being realized in R n in a natural way.LetΩbe a convex subset of R n,containing at least one non-zero point of P,and satisfying the following0-star condition:v∈Ωimpliesλv∈Ωfor all λ>1.In this case we will callΩa0-star sets.Denote by L(Ω)the vectorsubspace in G, spanned by e x,x∈Ω.Lemma6.L(Ω)is a Lie subalgebra of G.Proof.If x,y∈P∩Ωthen x+y=2(12y).12y belongs toΩbecause of theconvexity and thus x+y∈Ωby the0-star condition.Lemma7.LetΩbe a0-star set,v∈Ωandλ>0.Then,if the setΩλ,v=λv+Ωcontains at least one non-zero point of P,it is a0-star set.Moreover,Ωλ,v⊂Ω.Proof.Clearly,Ωλ,v is convex.Further,if w∈Ωandγ>1,thenγ(w+λv)=γ(λ+1)(1λ+1v)belongs toΩby the same arguments as in Lemma6.This completes theproof.Lemma8.LetΩbe a0-star set,v∈Ωandλ>0.Then L(Ωλ,v)is an ideal of L(Ω). Proof.If w∈Ωand w =w +v∈Ωλ,v for some w ∈Ω,then w+w =w+w +v∈Ωλ,v. This implies the statement.11Hence,for arbitrary G,Ωand v as above we can form the algebra A(P,ψ,Ω,λ,v)= L(Ω)/L(Ωλ,v),which we will call a Lie algebra of convex quasicrystal type.To obtain the usual Lie algebra of quasicrystal type,one should takeΩto be a half-space(open or closed), which does not contain0as an inner point.The basic properties of Lie algebras of convex quasicrystal type are similar to those of Lie algebra of quasicrystal type,however,their formulation is much more complicated because it usually depends on the structure ofΩ. Here we list only some most straightforward ones.Proposition4.Let A=A(P,ψ,Ω,λ,v)be a Lie algebra of convex quasicrystal type and a denote the inﬁnum of distances from points inΩ∩P to0.Assume that dim(A)>1and that for any x∈Ωsome neighborhood(in R n)of2x belongs toΩ.Then1.if a>0then any element of A is nilpotent.2.Anyﬁnite set of elements from A generates aﬁnite-dimensional Lie subalgebra of A. Proof.If x∈S=Ω\Ωλ,v then there alway exists y,such that|x−y| |v|and such that y∈Ω.Let w∈Ω.If some ball of radius r over2w belongs toΩ,then,forλ>1the point 2λw belongs toΩtogether with the ball of radiusλr around it.Makingλr>|v|we get that2λw∈Ωλ,v.This implies theﬁrst statement.If the set{w1,...,w k}⊂Ωisﬁnite,then weﬁnd some r such that2w i belongs toΩtogether with its neighbor ball of radius r.Then the same is true for all linear combinations of these elements with non-negative integer coeﬃcients.By the same arguments as in the previous paragraph,there is N∈N such that any linear combination of{Nw1,...,Nw k} with non-negative integer coeﬃcients belongs toΩλ,v.This implies the second statement.Let us study an example of such algebra,which,as we will show,has some interesting properties.Take P=Z2,ψthe projection on the second component,Ω={w∈R2: (w,(1,1)) 0and(w,(1,−1)) 0},v=(n+ ,0),n∈N, ∈(0,1).The corresponding algebra A=A(P,ψ,Ω,λ,v)is graded with respect to the e0action with graded components corresponding to all integers and having dimension n.In particular,one can deﬁne and study triangular(parabolic)decompositions of this algebra and corresponding(generalized) Verma modules.The set P =P∩(Ω\Ωλ,v)coincides with{(a,b):0 a−|b| n}. Deﬁne the conjugationωon this set viaω(a,b)=(2|b|+n−a,−b).Then we have the natural notions of Verma modules and the Shapovalov form on them.In our situation we have∆+=N.Lemma9.The Verma module M(Vλ)is always reducible.However,the unique sim-ple quotient of M(Vλ)is inﬁnite dimensional if and only if at least one of the numbers λ(e(2,0)),...,λ(e(n,0))is non-zero.Otherwise it is one-dimensional.Proof.We note that the intersection of[A,A]with A0coincides with the linear span˜A0 of elements{e(2,0),...,e(n,0)}.Hence,if the restriction ofλon˜A0is zero,the Shapovalov form in identically zero on all M(Vλ)λ−k,k∈N.Otherwise,assume thatλ(e(i,o))=0and take e x∈A1and e y∈A−1such that[e x,e y]= e(i,0).We get Fλ,k(e k,e k x)=0and the statement is proved.ω(y)12。

Field quantization in inhomogeneous anisotropic dielectrics with spatio-temporal dispersion

a r X i v :q u a n t -p h /0702010v 2 16 F eb 2007Field quantization in inhomogeneous anisotropic dielectrics with spatio-temporal dispersionL G Suttorp Instituut voor Theoretische Fysica,Universiteit van Amsterdam,Valckenierstraat 65,1018XE Amsterdam,The Netherlands Abstract.A quantum damped-polariton model is constructed for an inhomogeneous anisotropic linear dielectric with arbitrary dispersion in space and time.The model Hamiltonian is completely diagonalized by determining the creation and annihilation operators for the fundamental polariton modes as speciﬁc linear combinations of the basic dynamical variables.Explicit expressions are derived for the time-dependent operators describing the electromagnetic ﬁeld,the dielectric polarization and the noise term in the latter.It is shown how to identify bath variables that generate the dissipative dynamics of the medium.PACS numbers:42.50.Nn,71.36.+c,03.70.+k Submitted to:J.Phys.A:Math.Gen.1.IntroductionQuantization of the electromagneticﬁeld in a linear dielectric medium is a nontrivial task for various reasons.First of all,since the response of a dielectric to externalﬁelds is frequency-dependent in general,temporal dispersion should be taken into account.The well-known Kramers-Kronig relation implies that dispersion is necessarily accompanied by dissipation,so that the quantization procedure has to describe an electromagneticﬁeld that is subject to damping.Furthermore,since the transverse and the longitudinal parts of the electromagneticﬁeld play a diﬀerent role in the dynamics,the quantization scheme should treat these parts separately. For inhomogeneous and spatially dispersive media this leads to complications in the quantization procedure,which further increase in the presence of anisotropy.When the losses in a speciﬁc range of frequencies are small,temporal dispersion can be neglected.Field quantization in an inhomogeneous isotropic dielectric medium without spatio-temporal dispersion has been accomplished by employing a generalized transverse gauge,which depends on the dielectric constant[1]–[6].A phenomenological scheme forﬁeld quantization in lossy dielectrics has been formulated on the basis of theﬂuctuation-dissipation theorem[7]–[10].By adding a ﬂuctuating noise term to the Maxwell equations and postulating speciﬁc commutation relations for the operator associated with the noise,one arrives at a quantization procedure that has been quite successful in describing the electromagneticﬁeld in lossy dielectrics.An equivalent description in terms of auxiliaryﬁelds has been given as well[11,12],while a related formalism has been presented recently[13].However, all of these quantization schemes have the drawback that the precise physical nature of the noise term is not obvious,since its connection to the basic dynamical variables of the system is left unspeciﬁed.As a consequence,the status of the commutation relations for the noise operator is that of a postulate.A justiﬁcation of the above phenomenological quantization scheme has been sought by adopting a suitable model for lossy dielectrics.To that end use has been made of an extended version of the Hopﬁeld polariton model[14]in which damping eﬀects are accounted for by adding a dynamical coupling to a bath environment. Huttner and Barnett[15,16]were theﬁrst to employ such a damped-polariton model in order to achieveﬁeld quantization for a lossy dielectric.Their treatment,which is conﬁned to a spatially homogeneous medium,yields an explicit expression for the noise term as a linear combination of the canonical variables of the model.In a later development,an alternative formulation of the quantization procedure in terms of path integrals has been given[17],while Laplace transformations have been used to simplify the original formalism[18].More recently,the eﬀects of spatial inhomogeneities in the medium have been incorporated by solving an inhomogeneous version of the damped-polariton model[19]–[21].In this way a full understanding of the phenomenological quantization scheme has been reached,at least for those dielectrics that can be represented by the damped-polariton models mentioned above.The latter proviso implies a limitation in various ways.First,one would like to include in a general model not only the eﬀects of spatial inhomogeneity,but also those of spatial dispersion.Furthermore,it would be desirable to incorporate the consequences of spatial anisotropy,so that the theory encompasses crystalline media as well.Finally,while treating temporal dispersion and the associated damping,we would like to refrain from introducing a bath environment in the Hamiltonian from the start.Instead,we wish to formulate the Hamiltonian interms of a full set of material variables,from which the dielectric polarization emerges by a suitable projection.In this way we will be able to account for any temporal dispersion that is compatible with a few fundamental principles like causality and net dielectric loss.For a homogeneous isotropic dielectric without spatial dispersion such an approach has been suggested before[16,22].Recently,several attempts have been made to remove some of the limitations that are inherent to the earlier treatments.In[23]the eﬀects of spatial dispersion are considered in a path-integral formalism for a model that is a generalization of that of the original Huttner-Barnett approach.The discussion is conﬁned to homogeneous dielectrics and to leading orders in the wavenumber,so that an analysis of the eﬀects of arbitrary spatial dispersion in an inhomogeneous medium is out of reach.In [24]crystalline media have been discussed in the framework of a damped-polariton model with an anisotropic tensorial bath coupling.A complete diagonalization of the model along the lines of[15,16]turned out to face diﬃculties due to the tensorial complexity,so that the full dynamics of the model is not presented.Both spatial dispersion and anisotropy are incorporated in the quantization scheme discussed in [25].Use is made of a Langevin approach in which a damping term of a speciﬁc form is introduced.The commutation relations for the noise operator are postulated,as in the phenomenological quantization scheme.Finally,several treatments have appeared in which a dielectric model is formulated while avoiding the explicit introduction of a bath[26,27].However,a complete expression for the noise polarization operator in terms of the basic dynamical variables of the model is not presented in these papers.A direct proof of the algebraic properties of the latter operator is not furnished either.In the present paper,we shall show how the damped-polariton model can be generalized in such a way that all of the above restrictions are removed.As we shall see,our general model describes the quantization and the time evolution of the electromagneticﬁeld in an inhomogeneous anisotropic lossy dielectric with arbitrary spatio-temporal dispersion.A crucial step in arriving at our goals will be the complete diagonalization of the Hamiltonian.It will lead to explicit expressions for the operators describing the electromagneticﬁeld and the dielectric polarization,and for the noise contribution contained in the latter.In this way the commutation relations for the noise operator will be derived rigorously from our general model,instead of being postulated along the lines of the phenomenological scheme.Finally,we shall make contact with previous treatments by showing how to construct a bath that generates damping phenomena in the dynamical evolution of the model.2.Model HamiltonianIn this section we shall construct the general form of the Hamiltonian for a polariton model describing an anisotropic inhomogeneous dispersive dielectric.The result, which we shall obtain by starting from a few general principles,will contain several coeﬃcients that can be chosen at will.As we shall see in a subsequent section,these coeﬃcients can be adjusted in such a way that the susceptibility gets the appropriate form for any causal lossy dielectric that we would like to describe.The Hamiltonian of the electromagneticﬁeld is taken to have the standard form:H f= d r 12µ0[∇∧A(r)]2 (1) with the Hermitian vector potential A(r)and its associated Hermitian canonicalmomentumΠ(r).We use the Coulomb gauge∇·A=0.In this gauge bothΠand A are transverse.The canonical commutation relations read[Π(r),A(r′)]=−i¯hδT(r−r′),[Π(r),Π(r′)]=0,[A(r),A(r′)]=0(2) where the transverse delta function is deﬁned asδT(r)=Iδ(r)+∇∇(4πr)−1,with I the unit tensor.The Hamiltonian of the dielectric material medium is supposed to have the general formH m=¯h d r ∞0dωωC†m(r,ω)·C m(r,ω)(3) with the standard commutation relations for the creation and annihilation operators: [C m(r,ω),C†m(r′,ω′)]=Iδ(r−r′)δ(ω−ω′),[C m(r,ω),C m(r′,ω′)]=0.(4) The medium operators commute with theﬁeld operators.The material creation and annihilation operators are assumed to form a complete set describing all material degrees of freedom.Hence,any material dynamical variable, for instance the dielectric polarization density,can be expressed in terms of these operators.For a linear dielectric medium,the Hermitian polarization density is a linear combination of the medium operators,which has the general form:P(r)=−i¯h d r′ ∞0dω′C m(r′,ω′)·T(r′,r,ω′)+h.c.(5) The complex tensorial coeﬃcient T appearing in this expression will be determined later on,when the dielectric susceptibility is properly identiﬁed.On a par with P we deﬁne its associated canonical momentum density W,again as a linear combination of the medium operatorsW(r)=− d r′ ∞0dω′ω′C m(r′,ω′)·S(r′,r,ω′)+h.c.(6) with a new complex tensorial coeﬃcient S that is closely related to T,as we shall see below.For future convenience we inserted a factorω′in the integrand and a minus sign in front of the integral.As W and P are a canonical pair,they must satisfy the standard commutation relations[W(r),P(r′)]=−i¯h Iδ(r−r′),[W(r),W(r′)]=0,[P(r),P(r′)]=0.(7)Hence,the coeﬃcients S and T have to fulﬁll the requirements:d r′′ ∞0dω′′˜T(r′′,r,ω′′)·T∗(r′′,r′,ω′′)−c.c.=0(8)d r′′ ∞0dω′′ω′′˜S(r′′,r,ω′′)·T∗(r′′,r′,ω′′)+c.c.=Iδ(r−r′)(9)d r′′ ∞0dω′′ω′′2˜S(r′′,r,ω′′)·S∗(r′′,r′,ω′′)−c.c.=0(10)where the tilde denotes the transpose of a tensor and the asterisk the complex conjugate.Furthermore,the Hamiltonian should contain terms describing the interaction between theﬁeld and the medium.Two contributions can be distinguished:a transverse part and a longitudinal part.In a minimal-coupling scheme,which we shall adopt here,the transverse part is a bilinear expression involving the transversevector potential A and the canonical momentum density W.To ensure compatibility with Maxwell’s equations an expression quadratic in A should be present as well,as we shall see in the following.For dielectrics with spatial dispersion both expressions are non-local.The general form of the transverse contribution to the interaction Hamiltonian isH i=−¯h d r d r′W(r)·F1(r,r′)·A(r′)+1d r d r′∇·P(r)∇′·P(r′)2ε0 d r{[P(r)]L}2=Π(r,t)(13)ε0˙Π(r,t)=1ε0 d r′T∗(r,r′,ω)·[P(r′,t)]L′(15) where all operators now depend on time.The subscript L′denotes the longitudinal part with respect to r′.The time derivative of the polarization density follows by combining(5)and(15):˙P(r,t)=−¯h d r′ ∞0dω′ω′C m(r′,ω′,t)·T(r′,r,ω′)−¯h d r′ d r′′ d r′′′ ∞0dω′ω′˜T(r′,r,ω′)·S∗(r′,r′′,ω′)·F1(r′′,r′′′)·A(r′′′,t)¯h−iEliminatingΠfrom(13)and(14)weﬁnd an inhomogeneous wave equation for the vector potential∆A(r,t)−12ε0[Π(r)]2+12¯h d r d r′A(r)·F(r,r′)·A(r′)+1The complex tensorial coeﬃcient T can be chosen freely.It has to satisfy two constraints,theﬁrst of which has been written already in(8).The second one follows by substituting(20)in(10):d r′′ ∞0dω′′ω′′2˜T(r′′,r,ω′′)·T∗(r′′,r′,ω′′)−c.c.=0.(22)Finally,insertion of(20)in(9)leads to the equalityd r′′ ∞0dω′′ω′′˜T(r′′,r,ω′′)·T∗(r′′,r′,ω′′)+c.c.=F(r,r′).(23)This relation deﬁnes the real tensor F in terms of T.It shows that F(r,r′)satisﬁes the symmetry property˜F(r,r′)=F(r′,r),as we know already from the way F2occurs in (11).As an integral kernel the tensor F(r,r′)is positive-deﬁnite.This is established by taking the scalar products of(23)with real vectors v(r)and v(r′),and integrating over r and r′.The result is positive for any choice of v.As a consequence,the inverse of F is well deﬁned.The polarization density is given by(5),while the canonical momentum density reads according to(6)with(20):W(r)=− d r′ d r′′ ∞0dω′ω′C m(r′,ω′)·T(r′,r′′,ω′)·F−1(r′′,r)+h.c.(24) where the right-hand side contains the inverse of F.The Hamiltonian(21)has been constructed by starting from general forms for its parts H f,H m,H i and H es and requiring consistency with Maxwell’s equations.It may be related to a Lagrange formalism,as is shown in Appendix A.In the following we shall investigate the dynamics of the model deﬁned by(21). As the Hamiltonian is quadratic in the dynamical variables it is possible to accomplish a complete diagonalization.This will be the subject of the next section.3.Diagonalization of the HamiltonianWe wish toﬁnd a diagonal representation of the Hamiltonian(21)in the formH=¯h d r ∞0dωωC†(r,ω)·C(r,ω).(25) The creation and annihilation operators satisfy the standard commutation relations of the form(4).They are linear combinations of the dynamical variables in(21): C(r,ω)= d r′ f1(r,r′,ω)·A(r′)+f2(r,r′,ω)·Π(r′)+ ∞0dω′ f3(r,r′,ω,ω′)·C m(r′,ω′)+f4(r,r′,ω,ω′)·C†m(r′,ω′) (26) with as-yet unknown tensorial coeﬃcients f i,theﬁrst two of which are taken to be transverse in their second argument.To determine f i we use Fano’s method[28]:we evaluate the commutator[C(r,ω),H]and equate the result to¯hωC(r,ω).Comparing the contributions involving the various canonical operators we arrive at the four equationsii ε0 d r ′′ d r ′′′ ∞0dω′′{f 3(r ,r ′′,ω,ω′′)·[T ∗(r ′′,r ′′′,ω′′)]L ′′′+f 4(r ,r ′′,ω,ω′′)·[T (r ′′,r ′′′,ω′′)]L ′′′}·˜T (r ′,r ′′′,ω′)=ωf 3(r ,r ′,ω,ω′)(29)−i ¯h ω′ d r ′′f 2(r ,r ′′,ω)·˜T∗(r ′,r ′′,ω′)−ω′f 4(r ,r ′,ω,ω′)−¯h c 2d r ′′T ∗(r ,r ′′,ω)·[G (r ′′,r ′,ω−i 0)]T ′(31)f 2(r ,r ′,ω)=i µ0ω d r ′′T ∗(r ,r ′′,ω)·[G (r ′′,r ′,ω−i 0)]T ′(32)f 3(r ,r ′,ω,ω′)=I δ(r −r ′)δ(ω−ω′)−µ0¯h ω d r ′′ d r ′′′T ∗(r ,r ′′,ω)·[G (r ′′,r ′′′,ω−i 0)]T ′′′·˜T(r ′,r ′′′,ω′)+µ0¯h ω2ω+ω′d r ′′ d r ′′′T ∗(r ,r ′′,ω)·G (r ′′,r ′′′,ω−i 0)·˜T ∗(r ′,r ′′′,ω′).(34)The Green function G (r ,r ′,z )occurring in these expressions is deﬁned as the solution of the diﬀerential equation:− G (r ,r ′,z )×←−∇′ ×←−∇′+z 2c2 d r ′′G (r ,r ′′,z )·χ(r ′′,r ′,z )=I δ(r −r ′)(35)The spatial derivative operator ∇′acts to the left on the argument r ′of G (r ,r ′,z ).According to this inhomogeneous wave equation the Green function determines the propagation of waves through a medium that is characterized by a tensor χ(r ,r ′,z).The latter plays the role of a non-local anisotropic susceptibility,as will become clear in the next section.It is deﬁned in terms of T and its complex conjugate asχ(r,r′,z)≡¯hω−z˜T(r′′,r,ω)·T∗(r′′,r′,ω)+1ε0 d r′′˜T(r′′,r,ω)·T∗(r′′,r′,ω)(39) for positiveωand byχ(r,r′,ω+i0)−χ(r,r′,ω−i0)=−2πi¯h2πi ∞−∞dω1ε0F(r,r′)(43)∞−∞dωω2[χ(r,r′,ω+i0)−χ(r,r′,ω−i0)]=0.(44)Incidentally,we remark that for large|z|the asymptotic behaviour ofχfollows from (41)with(42)–(44)asχ(r,r′,z)≃−¯hz2+O 1c2G(r,r′,z)+ z2The medium operator C m (r ,ω)is a linear combination of the diagonalizing operator and its Hermitian conjugate:C m (r ,ω)= d r ′ ∞0dω′ ˜f ∗3(r ′,r ,ω′,ω)·C (r ′,ω′)−˜f 4(r ′,r ,ω′,ω)·C †(r ′,ω′) (53)asfollowsbytaking the inverse of (26).Substituting (33)–(34)and inserting the result in (5)we get after some algebraP (r ,t )=i ¯h c 2 d r′ d r ′′ ∞0dωω2[G (r ,r ′,ω+i 0)]L ·˜T(r ′′,r ′,ω)·C (r ′′,ω)e −i ωt +h .c .(55)From the Maxwell equation ∇·(ε0E +P )=0it follows that the left-hand side is proportional to the longitudinal part [E (r ,t )]L of the electric ﬁeld.The ensuing expression for the latter is analogous to (52),so that we arrive at the following result for the complete electric ﬁeld:E (r ,t )=i µ0¯h d r ′ d r ′′ ∞0dωω2G (r ,r ′,ω+i 0)·˜T(r ′′,r ′,ω)·C (r ′′,ω)e −i ωt +h .c .(56)Inspection of (54)shows that the polarization consists of two terms.The ﬁrst term is proportional to the electric ﬁeld,at least in Fourier space and after taking a spatial convolution integral.The proportionality factor is χ(r ,r ′,ω),which plays the role of a susceptibility tensor,as we anticipated in the previous section.The second term in (54)is not related to the electric ﬁeld.It represents a noise polarization density P n (r ,t )deﬁned as P n (r ,t )=−i ¯h d r ′ ∞0dω˜T (r ′,r ,ω)·C (r ′,ω)e −i ωt +h .c .(57)that has to be present so as to yield a quantization scheme in which the validity of the canonical commutation relations in the presence of dissipation is guaranteed.Introducing the Fourier transform P n (r ,ω)via P n (r ,t )= ∞dωP n (r ,ω)e −i ωt +h .c .(58)and its counterparts E (r ,ω)and P (r ,ω),we get from (54)with (56):P (r ,ω)= d r ′χ(r ,r ′,ω+i 0)·E (r ′,ω)+P n (r ,ω).(59)The Fourier-transformed noise polarization density is proportional to the diagonalizing operator:P n (r ,ω)=−i ¯h d r ′˜T(r ′,r ,ω)·C (r ′,ω).(60)as follows from (57)and(58).As wehavegot now an explicit expression for P n (r ,ω)we can derive its commutation relation.By employing (39)we obtain:P n (r ,ω),P †n(r ′,ω′) =−i ¯h ε0c 2 d r′ d r ′′ ∞0dωω2χ(r ,r ′,ω+i 0)·G (r ′,r ′′,ω+i 0)·P n (r ′′,ω)e −i ωt+ ∞0dωP n (r ,ω)e −i ωt +h .c .(64)By adding (62)and (64)we get an expression for the dielectric displacement D (r ,t ).Upon using (47)we may write it as D (r ,t )=− d r ′ ∞0dω∇×[∇×G (r ,r ′,ω+i 0)]·P n (r ′,ω)e −i ωt +h .c .(65)Clearly,the dielectric displacement is purely paring with (63)we ﬁnd that Maxwell’s equation ∇×B (r ,t )=µ0∂D (r ,t )/∂t is satisﬁed.It is instructive to return to the time-dependent representation of the linear constitutive relation (59):P (r ,t )= d r ′ t−∞dt ′χ(r ,r ′,t −t ′)·E (r ′,t ′)+P n (r ,t )(66)with the time-dependent susceptibility tensor deﬁned by writing:χ(r ,r ′,ω+i 0)= ∞0dt χ(r ,r ′,t )e i ωt .(67)The convolution integral in the ﬁrst term of (66),which expresses the causal response of the medium,depends on the electric ﬁeld at all times t ′preceding t and at all positions r ′,whereas the second contribution is the noise term,which in classical theory does not appear.Sometimes [26]a diﬀerent splitting of the various contributions to the polarization density is proposed,by writing an equation of the general form of (66)in which the response term covers only a limited range of values of t ′,for instance t ′∈[0,t ]for t >0.In such a formulation the convolution integral does not represent the full causal response of the medium,so that part of the response is hidden in the second term.As a consequence,the latter is no longer a pure noise term,so that it cannot be omitted in the classical version of the theory.The above expressions for the ﬁelds and the polarization density in terms of the Fourier-transformed noise polarization density satisfying the commutation relations (61)are the central results in the present formalism for ﬁeld quantization in inhomogeneous anisotropic dielectric media with spatio-temporal dispersion.Although we are describing dissipative media,it has not been necessary to explicitlyintroduce a bath,as is commonly done in the context of damped-polariton treatments [15,16,19,20].In the next section,we shall show how a bath may be identiﬁed in the present model.5.Bath degrees of freedomIn the Hamiltonian(21)the dielectric medium is described by the operators C m(r,ω) and C†m(r,ω).The polarization density P(r)and its canonical conjugate W(r)are given in(5)and(24)as suitable linear combinations of the medium operators C m and C†m.Since the latter depend on the continuous variableω,they describe many more degrees of freedom than P and W.The extra degrees of freedom can be taken together to deﬁne a so-called‘bath’,which is independent of P and W.Although the name might suggest otherwise,the bath as introduced in this way is part of the medium itself,and not some external environment.Its role is to account for the dissipative eﬀects in the dispersive medium,which may arise for instance through a leak of energy by heat production.In the following we shall identify the operators associated to the bath.Subsequently,we shall show how the Hamiltonian can be rewritten so as to give an explicit description of the coupling between the polarization and the bath.In this way,we will be able to compare our model to its counterparts in previous papers [15,16,19,20].The bath will be described by operators C b(r,ω)and C†b (r,ω)satisfying the usualcommutation relations.These bath operators are linear combinations of the medium operators:C b(r,ω)= d r′ ∞0dω′ H1(r,r′,ω,ω′)·C m(r′,ω′)+H2(r,r′,ω,ω′)·C†m(r′,ω′) (68) with tensor coeﬃcients H i that will be determined presently.Since the bath variables are by deﬁnition independent of both P(r′)and W(r′)for all r′,they have to commute with the latter.With the use of(5)and(24)we get from these commutation relations the following conditions:d r′′ ∞0dω′′[H1(r,r′′,ω,ω′′)·T∗(r′′,r′,ω′′)+H2(r,r′′,ω,ω′′)·T(r′′,r′,ω′′)]=0(69)d r′′ ∞0ω′′ω′′[H1(r,r′′,ω,ω′′)·T∗(r′′,r′,ω′′)−H2(r,r′′,ω,ω′′)·T(r′′,r′,ω′′)]=0.(70)To determine H i we start from the following Ansatz:H1(r,r′,ω,ω′)= d r′′ δ(ω−ω′)h1(r,r′′,ω)+1ω+ω′h2(r,r′′,ω)·˜T∗(r′,r′′,ω′)(72) with new tensor coeﬃcients h i.Substituting these expressions in(69)–(70)and using (36)and(39),weﬁnd that both of these conditions are simultaneously satisﬁed whenh 1and h 2are related as d r ′′h 2(r ,r ′′,ω)·χ(r ′′,r ′,ω+i 0)==12i d r ′′ d r ′′′h 1(r ,r ′′,ω)·[χ(r ′′,r ′′′,ω+i 0)−χ(r ′′,r ′′′,ω−i 0)]·˜h ∗1(r ′,r ′′′,ω)==π¯h ε0d r ′′T ∗(r ,r ′′,ω)·χ−1(r ′′,r ′,ω+i 0).(76)It should be noted that the coeﬃcients h i are determined up to a unitary transformation.This freedom,which is available to H i as well,corresponds to a natural arbitrariness in the choice of the bath operators themselves.As the bath operators have been identiﬁed now,we can rewrite the Hamiltonian so as to clarify their role in the dynamics of our model.To that end we have to eliminate the medium operators C m in favor of the bath operators C b .Employing(5),(24)and (68)we can write the medium operators as:C m (r ,ω)= d r ′T ∗(r ,r ′,ω)· i 2ε0[Π(r )]2+12πi ¯h 2 d r d r ′ d r ′′ d r ′′′P (r )·F −1(r ,r ′)· ∞0dωω3[χ(r ′,r ′′,ω+i 0)−χ(r ′,r ′′,ω−i 0)] ·F −1(r ′′,r ′′′)·P (r ′′′)+12¯hd r d r ′W (r )·F (r ,r ′)·W (r ′)−¯h d r d r ′W (r )·F (r ,r ′)·A (r ′)+1i¯h −causality and positive-deﬁniteness of the dissipative energy loss.Incidentally,it may be remarked that amplifying dielectric media,which have been treated in the context of the phenomenological quantization scheme as well[10,29],are not covered by the present damped-polariton model.To describe media with a sustained gain,e.g.a laser above threshold,one has to incorporate a driving mechanism in the Hamiltonian, which accounts for the ongoing input of energy that is indispensable for a stationary gain.As we have shown,the time evolution of the dynamical variables forﬁeld and matter can be determined completely by deriving the operators that diagonalize the Hamiltonian.The diagonalizing operators are closely related to the noise part of the polarization density,which plays an important role in the phenomenological quantization scheme.The proof of the commutation properties of the noise polarization density follows from its relation to the diagonalizing operators.In setting up our model Hamiltonian we have avoided to introduce a bath environment from the beginning.The subsequent formalism could be developed without ever discussing such a bath.Nevertheless,one may be interested in an analysis of the complete set of degrees of freedom of the dielectric medium in our model.If that analysis is carried out,oneﬁnds,as we have seen above,that speciﬁc combinations of medium variables can be associated to what may be called a bath.The coupling of the polarization to this bath can be held responsible for the dissipative losses that characterize a dispersive dielectric.AcknowledgmentsI would like to thank dr.A.J.van Wonderen for numerous discussions and critical comments.Appendix grangian formulationIn this appendix we shall show how the Hamiltonian(21)can be related to a Lagrange formalism.We start by postulating the following Lagrangian for an anisotropic linear dielectric with spatio-temporal dispersion that interacts with the electromagneticﬁeld: L= d r 12µ0[∇∧A(r)]2+12ε0 d r{[P(r)]L}2.(A.1) Here A(r)is the transverse vector potential and Q m(r,ω)are material coordinates depending on position and frequency.The polarization density P(r)is taken to be an anisotropic and non-local linear combination of these material coordinates of the form P(r)= d r′ ∞0dω′Q m(r′,ω′)·T0(r′,r,ω′)(A.2) with a real tensor coeﬃcient T0(r,r′,ω).One easily veriﬁes that the Lagrangian equations have the form1∆A(r,t)−¨Q m (r ,ω,t )+ω2Q m (r ,ω,t )= d r T 0(r ,r ′,ω)·E (r ′,t )(A.4)with the electric ﬁeld given as E (r ,t )=−˙A (r ,t )−(1/ε0)[P (r ,t )]L .The ﬁrst Lagrangian diﬀerential equation is consistent with Maxwell’s equation,as it should.The second Lagrangian equation shows that the material coordinates are harmonic variables that are driven by the electric ﬁeld in an anisotropic and non-local way.Introducing the momenta Π(r )and P m (r ,ω)associated to A and Q m asΠ(r )=ε0˙A (r )(A.5)P m (r ,ω)=˙Q m (r ,ω)+ d r ′T 0(r ,r ′,ω)·A (r ′)(A.6)we obtain the Hamiltonian corresponding to (A.1)in the standard fashion.The result is:H = d r 12µ0[∇∧A (r )]2+12 d r d r ′ d r ′′ ∞dωA (r )·˜T 0(r ′,r ,ω)·T 0(r ′,r ′′,ω)·A (r ′′)+12 1/2d r ′ C m (r ′,ω)·U (r ′,r ,ω)+C †m (r ′,ω)·U ∗(r ′,r ,ω) (A.8)Q m (r ,ω)=i ¯hAppendix B.Evaluation of the tensorial coeﬃcients f i In thisappendix wewill showhowtheequations (27-30)can be solved.We start by using (27)to eliminate f 1from (28).As a result we obtain the diﬀerential equation:∆′f 2(r ,r ′,ω)+ω2ε0 d r′′ d r ′′′ d r ′′′′ ∞0dω′′{f 3(r ,r ′′,ω,ω′′)·[T ∗(r ′′,r ′′′,ω′′)]L ′′′+f 4(r ,r ′′,ω,ω′′)·[T (r ′′,r ′′′,ω′′)]L ′′′}·˜T(r ′′′′,r ′′′,ω′)·T ∗(r ′′′′,r ′,ω′)=0.(B.2)A similar relation is obtained by multiplying (30)by T (r ′,r ′′′′,ω′)and integrating over r ′:−i ¯h ω′ d r ′′ d r ′′′f 2(r ,r ′′,ω)·˜T∗(r ′′′,r ,′′,ω′)·T (r ′′′,r ′,ω′)−(ω+ω′) d r ′′f 4(r ,r ′′,ω,ω′)·T (r ′′,r ′,ω′)−¯h c 2f 2(r ,r ′,ω)−i µ0ωd r ′′ ∞0dω′′{f 3(r ,r ′′,ω,ω′′)·[T ∗(r ′′,r ′,ω′′)]T ′+f 4(r ,r ′′,ω,ω′′)·[T (r ′′,r ′,ω′′)]T ′}=0.(B.5)Here we used the transversality of f 2(r ,r ′,ω)in its second argument to write the ﬁrst term as a repeated vector product,with the spatial derivative operator ∇′acting to the left on theargument r ′of thefunction f 2(r ,r ′,ω).The integral in (B.5)contains the transverse parts of T and T ∗only.A more natural form of the diﬀerential equation,with the full tensors T and T ∗,is obtained by introducing instead of f 2a new tensor g deﬁned as:g (r ,r ′,ω)≡i ωf 2(r ,r ′,ω)−1c 2g (r ,r ′,ω)+µ0ω2d r ′′ ∞0dω′′[f 3(r ,r ′′,ω,ω′′)·T ∗(r ′′,r ′,ω′′)+f 4(r ,r ′′,ω,ω′′)·T (r ′′,r ′,ω′′)]=0.(B.7)The integral contribution still depends on f 3and f 4,so that the diﬀerential equation is not yet in closed form.However,we may rewrite the integral in such a way that its relation to g becomes obvious.This can be achieved with the help of the identity: d r ′′ ∞0dω′′[f 3(r ,r ′′,ω,ω′′)·T ∗(r ′′,r ′,ω′′)+f 4(r ,r ′′,ω,ω′′)·T (r ′′,r ′,ω′′)]==ε0 d r ′′g (r ,r ′′,ω)·χ(r ′′,r ′,ω−i 0)+s (r ,r ′,ω).(B.8)which contains a tensor s (r ,r ′,ω)that arises while avoiding a pole in the complex frequency plane,as we shall see below.Furthermore the right-hand side contains the susceptibility tensor χthat has been deﬁned in (36).In (B.8)the frequency is chosen to be in the lower half of the complex plane just below the real axis.Correspondingly,the term −i 0is an inﬁnitesimally small number on the negative imaginary axis.To prove (B.8)we divide (B.2)by ω′−ω+i 0,with i 0an inﬁnitesimally small imaginary number.The result is:−i ¯h ω′ε01ω+ω′ d r ′′ d r ′′′f 2(r ,r ′′,ω)·˜T∗(r ′′′,r ′′,ω′)·T (r ′′′,r ′,ω′)。

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a r X i v :m a t h /0603408v 1 [m a t h .C A ] 16 M a r 2006Symmetry,Integrability and Geometry:Methods and Applications Vol.2(2006),Paper 034,8pagesOn Orthogonality Relationsfor Dual Discrete q -Ultraspherical PolynomialsValentyna A.GROZA†and Ivan I.KACHURYK‡†National Aviation University,1Komarov Ave.,Kyiv,03058Ukraine E-mail:groza@.ua‡Khmel’nyts’kyi National University,Khmel’nyts’kyi,Ukraine E-mail:kachuryk@ief.tup.km.uaReceived February 14,2006,in final form February 28,2006;Published online March 16,2006Original article is available at http://www.emis.de/journals/SIGMA/2006/Paper034/Submitted by Anatoly KlimykAbstract.The dual discrete q -ultraspherical polynomials D (s )n (µ(x ;s )|q )correspond to in-determinate moment problem and,therefore,have one-parameter family of extremal ortho-gonality relations.It is shown that special cases of dual discrete q -ultraspherical polynomials D (s )n (µ(x ;s )|q ),when s =q −1and s =q ,are directly connected with q −1-Hermite polyno-mials.These connections are given in an explicit ing these relations,all extremalorthogonality relations for these special cases of polynomials D (s )n (µ(x ;s )|q )are found.Key words:q -orthogonal polynomials;dual discrete q -ultraspherical polynomials;q −1-Her-mite polynomials;orthogonality relation2000Mathematics Subject Classiﬁcation:33D45;81Q991IntroductionQuantum groups and q -deformed variants of the quantum harmonic oscillator were introduced approximately two decades ago.Representations of quantum groups and realizations of q -oscillators are closely related to basic hypergeometric functions and q -orthogonal polynomials.Instead of Jacobi,Gegenbauer (ultraspherical),Hermite polynomials,in the theory of quantum groups and under studying q -oscillators we deal with different kinds of q -Jacobi polynomials,of q -ultraspherical polynomials and of q -Hermite polynomials.It is known that q -orthogonal polynomials are more complicated than classical orthogonal polynomials.It is interesting that main development of the theory of q -orthogonal polynomials in fact coincides in time with appearing of quantum groups (see,for example,[1]and [2]).Due to great importance for contemporary mathematical and theoretical physics the theory of q -orthogonal polynomials is under extensive development.A great attention is paid to those of them,which are directly related to representations of quantum groups and to realizations of q -oscillators.In this paper we deal with the so-called q −1-Hermite polynomials (which are closely related to the Biedenharn–Macfarlane oscillator;see [3])and with the dual discrete q -ultraspherical polynomials.The q −1-Hermite orthogonal polynomials were discovered by Askey [4]and the discrete q -ultraspherical polynomials and their duals were introduced in [5].The q −1-Hermite orthogonal polynomials correspond to indeterminate moment problem.This means that they have a one-parameter family of orthogonality relations (these orthogonalities2V.A.Groza and I.I.Kachurykx n(b1;q)n(b2;q)n...(b r−1;q)n√sq,−√sq,−q k(2k−1)D(s)n(µ(2k)|q)D(s)n′(µ(2k)|q)(1−sq)(q;q)2kOn Orthogonality Relations for Dual Discrete q-Ultraspherical Polynomials3 (q;q2)∞(q2;q2)n q−n(1−sq)(q;q)2k+1q k(2k+1)D(s)n(µ(2k+1)|q)D(s)n′(µ(2k+1)|q)=(sq3;q2)∞(sq2;q2)nδnn′,(3)whereµ(2k)≡µ(2k;s),µ(2k+1)≡µ(2k+1;s)and0<s<q−2.We shall need also q−1-Hermite polynomials h n(x|q),which are given ash n(sinhϕ|q)=nk=0(−1)k q k(k−n)(q;q)n(−a2;q)∞(−q/a2;q)∞(q;q)∞h n(x m|q)h n′(x m|q)=q−n(n+1)/2(q;q)nδnn′,(5) wherex m=14V.A.Groza and I.I.KachurykOn Orthogonality Relations for Dual Discrete q-Ultraspherical Polynomials5(q;q2)∞(q2;q2)n q−n2(q−m−q m)|q)and substitute into(14).As a result,we obtain∞m=01+q2m q m(2m−1)h2n(x m|q)h2n′(x m|q)=(q2;q2)∞2(q−m−q m).Note that(q2;q2)∞2(−1;q)∞(−q;q)∞(q;q)∞=2q−1 −q2;q ∞ −q−1;q ∞(q;q)∞.(16)The expression(1+q2m)q m(2m−1)on the right hand side of(15)does not change at replacing m by−m.Since h2n(x|q)is an even function,then due to(16)the relation(15)can be written as ∞m=−∞(1+q2m)q−1q m(2m−1)6V.A.Groza and I.I.Kachuryk(q ;q 2)∞(1−q )2(q 2;q 2)n q −n2(q m−q −m )|q )and substitute into(18).We obtain∞m =11+q 2mq m (2m −1)h 2n +1(x m |q )h 2n ′+1(x m |q )=(q 2;q 2)∞2(q −m−q m ).Since h 2n +1(0|q )=0,then the summation ∞m =1here canbe replaced by∞m =0,and then by1q k (q ;q )−12k (q ;q 2)2kδkk ′,(21)wherey m =a −2q −2m +a 2q 2m .These relations are parametrized by a ,q ≤a <1.For different values of a the correspondingorthogonality relations are not equivalent.On Orthogonality Relations for Dual Discrete q -Ultraspherical Polynomials 7q k +1(q ;q )−12k +1(q 3;q 2)2kδkk ′(22)wherey m = a −2q −2m +a 2q 2mq.These relations are also parametrized by a ,q ≤a <1.Orthogonality relations,corresponding to different values of a ,are not equivalent.The orthogonality relations (5)for the q −1-Hermite polynomials are extremal,that is,the set of polynomials h n (x m |q ),n =0,1,2,...,is complete in the corresponding Hilbert space (see [11]).Therefore,the orthogonality relations (21)and (22)for the polynomials D (q −1)kand D (q )k ,respec-tively,are extremal.Moreover,since the set of orthogonality relations (5)is a complete set of extremal orthogonality relations for the q −1-Hermite polynomials,then the set of relations (21)is a complete set of extremal orthogonality relations for the polynomials D (q −1)kand the set of relations (22)is a complete set of extremal orthogonality relations for the polynomials D (q )k .5Concluding remarksThe dual discrete q -ultraspherical polynomials D (s )n (µ(x ;s )|q )correspond to the indeterminate moment problem,that is,there are infinitely many (one-parameter family)of orthogonality relations for them.We have found these orthogonality relations for the cases when s =q and s =q −ly,we have reduced these cases to the q −1-Hermite polynomials for which orthogonality relations are known.It leaves unsolved the problem of finding orthogonality relations for other cases.We believe that the orthogonality relations (21)and (22)will help in finding orthogonality relations for these other cases.There exists another class of dual discrete q -ultraspherical polynomials (they are denoted as ˜D (s )n (µ(x ;−s )|q );see [5,Section 1]).They also correspond to the indeterminate moment problem.It is interesting to find orthogonality relations for them.However,they cannot be reduced to the q −1-Hermite polynomials.It is shown in [5]that they are reduced to Berg–Ismail polynomials.This connection gives an infinite number of orthogonality relations for them.However,they are not extremal orthogonality relations.So,the problem of finding these orthogonality relations are unsolved.It is well-known (see [3])that q −1-Hermite polynomials are closely related to the Biedenharn–Macfarlane q -oscillator.Thus,the polynomials,considered in this paper,is also related to thisq -oscillator.Then the following question appears:Are the polynomials ˜D (s )n (µ(x ;−s )|q )relatedto the q -oscillator?Probably,they correspond to some simple deformation of the q -oscillator.8V.A.Groza and I.I.Kachuryk。