P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras

Commun.Theor.Phys.(Beijing,China)50(2008)pp.7–12cChinese Physical Society Vol.50,No.1,July 15,2008P -Twisted Affine Lie Algebras and Their Associated Vertex Algebras ∗DING Lu 1,†and ZHENG Zhu-Jun 2,3,‡1Institute of Applied Mathematics,Academy of Mathematics and Systems Science,the Chinese Academy of Sciences,Beijing 100080,China 2School of Mathematical Sciences,South China University of Technology,Guangzhou 510641,China3Institute of Mathematics,Henan University,Kaifeng 475001,China(Received July 26,2007;Revised October 22,2007)Abstract For any finite-dimensional semisimple Lie algebra g ,a Z +-graded vertex algebra is construsted on thevacuum representation V k (ˆg [θ])of ˆg [θ],which is a one-dimentional central extension of θ-invariant subspace on the loopalgebra L g =g ⊗C ((t 1/p)).PACS numbers:02.20.TwKey words:p -twisted affine Lie algebra,vertex algebra1IntroductionVertex operator algebras are a class of algebras,whose structures are arisen naturally from vertex operator con-structions of representations of affine Lie algebras and in the work of Frenkel–Lepowsky–Meurman and Borcherds on the “moonshine module”for the Monster finite sim-ple group.[1−5]They are (rigorous)mathematical coun-terparts of chiral algebras in 2-dimensional quantum con-formal field theory in physics.There are two differential approaches to vertex algebras,one is algebraic in which vertex algebras appear as formal power series acting on graded vector spaces.The other is geometric and more abstract —the theory of chiral algebras and factorization algebras developed by A.Berlinson and V.Drinfeld.[6]In papers [7]and [8],the second author and his coau-thors defined p -twisted affine Lie algebra.As examples,they constructed explicitly the 3-twisted affine Lie alge-bra sl (2,C )[θ]and 7-twisted affine Lie algebra sl (3,C )[θ].Furthermore,they constructed their vertex algebra repre-sentations.But,they did not give the associated vertex algebra structure.In the paper [9],the second author and his coauthor constructed a vertex algebra associated to sl (2,C )[θ].In this paper,a class of p -twisted affine Lie algebra ˆg [θ]and its vacuum representation is constructed.Based on the method of Frankel,[10]Y.Z.Huang [11]etc.,a new construction of vertex algebra associated to the p -twisted affine algebra ˆg [θ]is given.2p -Twisted Affine Lie Algebra ˆg [θ]Suppose g is a finite-semisimple-dimensional Lie alge-bra,p is a fixed prime number,and ω=exp(2π√−1/p ).Let θ1be a Lie algebra isomorphism on g,which satisfiesthat θp1=Id and its restriction on Carton subalgebra ηis identity.Splitting the Lie algebra g into a direct sum asg =p −1 j =0g j ,where g j ={A ∈g |θ1(A )=ωjA }.Consider the vector space C ((t 1/p ))of formal Laurent series in one variable as a commutative Lie algebra.Wecan define a Lie algebra automorphism θ2on C ((t 1/p ))sat-isfying θp2=Id and θ2(fg )=θ2(f )θ2(g )for any elements f ,g in C ((t 1/p )).Based on the isomorphism,a direct sum decomposition of C ((t 1/p ))can be got asC ((t 1/p))=p −1i =0C ((t 1/p ))(i ),where C ((t 1/p ))(i )={f (t )∈C ((t 1/p ))|θ2(f (t ))=ωi f (t )}.A formal loop algebra L g =g((t 1/p ))=g ⊗C ((t 1/p ))of g is defined as a Lie algebra with the commutator[A ⊗f (t ),B ⊗g (t )]=[A,B ]⊗(f (t )·g (t )).Using the automorphisms θ1and θ2,we may have an ex-tended automorphism on L g,θ:L g −→L g ,A ⊗f (t )−→θ1(A )⊗θ2(f (t )).Denote its invariant subspace by L g[θ].Clearly,it is a Lie algebra because for any element A ⊗f (t ),B ⊗g (t )in L g[θ],we haveθ([A ⊗f (t ),B ⊗g (t )])=θ([A,B ]⊗f (t )g (t ))=[θ1(A ),θ1(B )]⊗θ2(f (t ))θ2(g (t ))=[θ1(A )⊗θ2(f (t )),θ1(B )⊗θ2(g (t ))]=[θ(A ⊗f (t )),θ(B ⊗g (t ))]=[A ⊗f (t ),B ⊗g (t )].Choose homogeneous elements in g (j ),j ∈{0,...,p −1}as a basis of g,J a with a ∈I .Let −1 a be max {i ∈(1/p )Z −|J a ⊗t i ∈L g[θ]},and n a =n +1+ −1 a with n ∈Z .The elements J a ⊗t n a with a ∈I form a topo-logical basis for L g[θ].We denote J a ⊗t n a by J an a forsimplicity.If there is no confusion,we also write J an a as J a n and n a as n .Let (,)be the Killing form of g.We construct a p -twisted affine Lie algebra ˆg [θ]which is a one-dimensional central extension of L g[θ]by0−→C K −→ˆg [θ]−→L g[θ]−→0∗Theproject supported by National Key Basic Research Project of China under Grant Nos.2004CB318000and 2006CB805905,Na-tional Natural Science Foundation of China under Grant No.10471034,the Outstanding Youth Fund of Henan Province under Grant No.0512000100,and Innovation Fund of Colleges and Universities in Henan Province †E-mail:dinglu@ ‡E-mail:zhengzj@8DING Lu and ZHENG Zhu-Jun Vol.50with the Lie brackets [K,ˆg [θ]]=0and [J a n a ,J bm b ]=[J a ,J b ]⊗t n a + m b + n a δ n a ,− m b (J a ,J b )K .Let ˆg [θ]+=(g ⊗C [[t 1/p ]])∩L g[θ]and ˆg [θ]−=(g ⊗t −1/p C [t −1/p ])∩L g[θ].The affine Lie algebra ˆg [θ]can be decomposed asˆg [θ]=C K ⊕ˆg [θ]+⊕ˆg [θ]−.(1)Clearly,{J an a ,K |a ∈I,n ∈Z }is a topological basis ofˆg [θ],{J a n a |a ∈I,n ∈Z +}and {J am a |a ∈I,m ∈Z −}are topological bases of ˆg [θ]+and ˆg [θ]−respectively.3Vacuum Representation of ˆg[θ]Consider the one-dimensional representation C k ofˆg [θ]+⊕C K ,on which ˆg [θ]+acts by 0and K acts as mul-tiplication by a scalar k ∈C .We define the vacuum rep-resentation of level k of ˆg [θ]as the representation induced from C k ,V k (ˆg [θ])=Ind ˆg [θ]ˆg [θ]+⊕C K C k=U (ˆg [θ])U (ˆg [θ]+⊕C K )C k ,where U (ˆg [θ])and U (ˆg [θ]+⊕C K )are universal enveloping algebras of ˆg [θ]and ˆg [θ]+⊕C K respectively.By Eq.(1),we get an isomorphisms of vector spacesU (ˆg [θ])∼=U (ˆg [θ]−)⊗U (ˆg [θ]+⊕C K ).HenceV k (ˆg [θ])∼=U (ˆg [θ]−)⊗C k .Denote by |0 the image of 1⊗1∈U (ˆg [θ]−)⊗C kin V k (ˆg [θ]).By the Poincar´e –Birkhoff–Witt theorem,V k (ˆg [θ])has a PBW basis of monomials of the formJ a 1 −n 1 a 1···J a m −n m a m|0 ,(2)where m ∈Z +, −n 1 a 1≤ −n 2 a 2≤···≤ −n m a m <0.For case −n i a i = −n i +1 a i +1,we have J a i J a i +1,where i =1,2,...,m −1.4Vertex Algebra Structure on V k (ˆg [θ])In this section,a Z +-graded vertex algebra associ-ated to the vacuum representation V k (ˆg [θ])is constructed.First,the following data are given as (i)Vacuum vector |0 ;(ii)The translation operator T on V k (ˆg [θ])is definedby the rules T |0 =0and [T,J a −n a ]=nJ a−n −1 a ;(iii)Z +-gradation of V k (ˆg [θ])is defined as deg J a 1 −n 1 ···J a m−n m |0 =n 1+n 2+···+n m ,deg |0 =0.SinceT (J a 1 −n 1 ···J a m −n m |0 )=J a 1 −n 1 T J a 2 −n 2 ···J a m −n m |0 +n 1J a 1 −n 1−1 J a 2 −n 2 ···J a m−n m |0=J a 1−n 1···J a m−n m T |0+m i =1n i J a 1 −n 1 ···J a i −n i −1 ···J a m−n m |0=m i =1n i J a 1 −n 1 ···J a i −n i −1 ···J a m−n m |0 ,the action of T is uniquely determined on the PBW basis of V k (ˆg [θ]).Consequently,T is uniquely determined onV k (ˆg [θ]).Define the main ingredient of the vertex algebra structure —the vertex operator Y (.,z )on V k (ˆg [θ])asY (|0 ,z )=Id ,Y J a −1 |0 ,z = n ∈ZJ a n z −n −1,Y J a −n |0 ,z =1(n −1)!∂n −1zY J a −1|0 ,z ,Y J a 1 −n 1 ···J a m −n m |0 ,z =:Y J a 1 −n 1 |0 ,z ···Y J a m −n m |0 ,z :,(3)where :A (z )B (w ):is the normally ordered product of thefields A (z )and B (w ).Lemma 1Y J a −n |0 ,zis a field on V k (ˆg [θ])with n ∈N .Proof Note that for any a ∈I ,n ∈Z +,J an |0 =0.Fix a number m in Z +.Assume that if n is large enough we have the following formula J an J a 1 −n 1 ···J a m −n m|0 =0,(4)for any n i ∈N ,a,a i ∈I with i =1,...,m andJ an J a 1 −n 1 ···J a m +1 −n m +1 |0 = J a n ,J a 1 −n 1 +J a 1 −n 1 J a n J a 2 −n 2 ···J a m +1 −n m +1|0 = [J a ,J a 1] n a + −n 1 a 1+ n a δ n a ,− −n 1 a 1(J a ,J a 1)K× J a 2 −n 2 ···J a m +1 −n m +1|0 +J a 1 −n 1 J an J a 2 −n 2 ···J a m +1 −n m +1|0 ,when n 0,by Eq.(4),we haveJ a n J a 1 −n 1 ···J a m +1 −n m +1|0 =0,n 0.It implies that for any v ∈V k (ˆg [θ]),J an (v )=0,when n 0.(5)From Y J a−m|0 ,z =1(m −1)!∂m −1z Y J a −1|0 ,z =1(m −1)!n ∈Z(−n −1)···(−n −m +1)J a n z −n −m,we get the conclusion that Y (J a−m |0 ,z )is also a field.Since the normally ordered product of any two fields is also a field,we have Y J a 1 −n 1 ···J a m−n m|0 ,z =:Y J a 1 −n 1 |0 ,z ···Y J a m −n m|0 ,z :is a field.Hence,Y (A,z )is a field for any element A inV k (ˆg [θ])because J a 1 −n 1 ···J a m−n m |0 is a topological basis of V k (ˆg [θ]).Next we give the proof that the linear operators T and Y (.,z )satisfy vacuum axiom,translation axiom and lo-cality axiom.Firstly,checking the vacuum axiom,that is,lim z →0Y (A,z )|0 =A for any A ∈V k (ˆg [θ]).The state-ment Y (|0 ,z )|0 =|0 follows from our definition.SinceNo.1P -Twisted Affine Lie Algebras and Their Associated Vertex Algebras 9when n 0,J an acts by 0on |0 ,we obtain Y J a−m |0 ,z |0=1(m −1)!n ≤−m(−n −1)···(−n −m +1)J a n |0 z −n −m→J a −m |0 ,if z →0,(6)where m ≥1.Lemma 2If the vacuum axiom holds for the fieldY (A,z )=n ∈Z A n z −n −1,then it also holds for the fieldY (J a −s A,z )for any s ∈N .Proof By definition and some simple calculations,Y J a −sA,z=(s −1)!m ∈Z n ≤−s(−n −1)···(−n −s +1)J a n A m −n + n ≥0(−n −1)···(−n −s +1)A m −n J an z −s −m −1.Since the second term kills |0 for J a n |0=0with n ≥0,Y (J a−s A,z )|0is given by the power series with positivepowers of z only,with the constant term J a −s A −1|0 =J a−s A . By Lemma 2and Formula (6),the vacuum axiom is satisfied.Next we check the translation axiom.Firstly,[T,Y (|0 ,z )]=[T,Id ]=0.By [T,J a −n a ]=nJ a−n −1 a ,and the definition of the vertex operator Y (·,z )T,Y J a −1 |0 ,z = T, n ∈ZJ a n z −n −1 = n ∈ZT,J a n z −n −1=− n ∈Z(n +1)J a n z −n −2=∂z Y J a −1|0 ,z , T,Y J a −m|0 ,z =1(m −1)!∂(m −1)z T,Y J a −1 |0 ,z =1(m −1)!∂(m )zY (J a−1 |0 ,z )=∂z Y J a −m|0 ,z .Lemma 3Suppose A (z ),B (z )are fields on V ,and satisfy that [T,A (z )]=∂z A (z ),[T,B (z )]=∂z B (z ),then [T,:A (z )B (z ):]=∂z :A (z )B (z ):.Proof By definition,we have [T,A n ]=−nA n −1,[T,B n ]=−nB n −1.Let C s = n<0A n B s −n −1+n ≥0B s −n −1A n .By[T,C s ]= n<0[T,A n B s −n −1]+n ≥0[T,B s −n −1A n ]= n<0 [T,A n ]B s −n −1+A n [T,B s −n −1] + n ≥0[T,B s −n −1]A n +B s −n −1[T,A n ]=n<0(−nA n −1B s −n −1−(s −n −1)A n B s −n −2)+n ≥0(−(s −n −1)B s −n −2A n −nB s −n −1A n −1)=−n<0sA n B s −n −2−n ≥0sB s −n −2A n =−sC s −1,For A (z )B (z ):=s ∈Z C s z −s −1,[T,:A (z )B (z ):]= s ∈Z [T,C s ]z −s −1=− s ∈Z (s +1)C s z −s −2,hence [T,:A (z )B (z ):]=∂z :A (z )B (z ):. Since Y J a 1 −n 1 ···J a m−n m |0 ,z =:Y J a 1 −n 1 |0 ,z ···:Y J a m −1 −n m −1 |0 ,z Y J a m −n m |0 ,z ::,then by Lemma 3,we have T,Y J a 1 −n 1 ···J a m −n m |0 ,z =∂z Y J a 1 −n 1 ···J a m−n m|0 ,z .Hence the translation axiom is satisfied because [,]and Y (.,z )are linear.Finally,we check the locality axiom.Suppose a,b are two elements in I,and J a ∈g (i ),J b ∈g (j ).Let H = n,m ∈Z [J a ,J b ] n a + m b z −n −1w −m −1and L = n,m ∈Z n a (J a ,J b )δ n a ,− m b kz −n −1w −m −1.By calculations,we know that Y (J a−1 a |0 ,z ),Y J b −1 b |0 ,w ) =L +H .By the definition of · ,we getL = n,m ∈Z(n +1+ −1 a ) J a ,J b δ−(n +m +2), −1 a + −1 b kz −n −1w −m −1 .There are two cases.Case 1 −1 a + −1 b ∈Z .In this case,L =0is obvious.Case 2 −1 a + −1 b ∈Z .By computationL = n ∈Z(1+ −1 a ) J a ,J b kz −n −1w n +1+ −1 a + −1 b + n ∈Zn J a ,J b kz −n −1wn +1+ −1 a + −1 b=(1+ −1 a ) J a ,J b kw 1+ −1 a + −1 bδ(z −w )+ J a ,J bk∂w δ(z −w )w 2+ −1 a + −1 b .Hence (z −w )2L =0.Since θ1 J a ,J b= θ1(J a ),θ1(J b ) =ωi +j J a ,J b ,then J a ,J b ∈g (k )with k =i +j −pZ ab ,where Z ab = 0,i +j <p1,i +j ≥p.Hence there exists {J c i }which is a basis of g (k )and complex numbers C i such that J a ,J b =10DING Lu and ZHENG Zhu-Jun Vol.50dim g (k )i =1C i J c i .Noticing that there exists Z ab in Z such that −1 c i = −1 a + −1 b +Z ab ,and J a −1 a ,J b −1 b =dim g (k )i =1C i J c i ⊗t −1 a + −1 b ∈L g[θ],H = n,m ∈Zdim g (k )i =1C i J c in +m +2+ −1 ci−Z abz−n −1w−m −1=n,m ∈Zdim g (k )i =1C i J c i n +m +1−Z ab c iz−n −1w −m −1=n,m ∈Z dim g (k )i =1C i J c i n c iz−n +m −Z ab w −m −1=dim g (k )i =1C i Y J c i −1 ci|0 ,z m ∈Zz m −Z ab+1w −m −1=dim g (k )i =1C i Y J c i −1 ci|0 ,zδ(z −w )z −Z ab +1.For (z −w )δ(z −w )=0,(z −w )H =0.Now we get the conclusion (z −w )2 Y J a −1 a |0 ,z ,Y J b −1 b |0 ,w=0.Differentiating the formula with respect to z ,and multiplying the result by (z −w ),we obtain (z −w )3 Y J a −2 a |0 ,z ,Y J b −1 b|0 ,w =0.So Y J a −2 a |0 ,z and Y J b −1 b |0 ,w are mutually local.By induction,Y J a−n a |0 ,z ,Y J b −m b |0 ,w are local for any n,m ∈N ,and a,b ∈I .Using Dong’s Lemma,we get the locality of Y J a 1 −n 1 a 1···J a i −n i a i |0 ,z and Y J b 1 −m 1 b 1···J b j −m j b j|0 ,w .Therefore we get our main conclusion that (V k ˆg [θ]),|0 ,T,Y (.,z )is a vertex algebra.In fact,it is also a Z +-graded vertex algebra.This will be proved in the following.Lemma 4The degree of T is 1.Proof Noting that T J a−n |0 =J a −n T |0 +nJ a −n −1 |0 =nJ a −n −1 |0 ,and for any J a m −n m ···J a 1 −n 1 |0 ∈V k (ˆg [θ]),T J a m +1−n m +1 ···J a 1−n 1 |0 = T,J a m +1 −n m +1 +J a m +1 −n m +1 T J a m −n m ···J a 1 −n 1 |0=(n m +1)J a m +1 −n m +1−1 J a m−n m···J a 1 −n 1 |0 +J a m +1 −n m +1 T J a m−n m ···J a 1 −n 1 |0 ,deg T =1. Lemma 5For any A ∈V k (ˆg [θ])with deg A =m ,the field Y (A,z )has conformal dimension m .Proof The conformal dimension of Y J a −1 a|0 ,z = n ∈ZJ an a z −n −1is 1obtained directly from its definition.Since Y (J a −j a |0 ,z )=1(j −1)!∂j −1z Y J a −1 a |0 ,z = n ∈Z(−n −1)···(−n −j +1)(j −1)!J a n a z −n −j=n ∈Z(j −n −2)···(−n )(j −1)!J a n +1−j a z −n −1,and deg J an +1−j a =j −n −1,then the conformal dimen-sion is j .Hence Y J a m −n m ···J a 1−n 1|0 ,z =:Y t J a m −n m |0 ,z ···Y J a 1 −n 1|0 ,z:has the conformal dimension n m +···+n 1.SinceJ a m −n m ···J a 1 −n 1 |0 is a basis of V k (ˆg [θ])and Y (.,z )is linear,we obtain the result. By Lemma 4and Lemma 5, V k ˆg [θ] ,|0 ,T,Y (.,z )is a Z +-graded vertex algebra.5ExampleIn Subsec.5.1,we give some concrete constructions of the automorphism θin Lie algebra sl (n,C )and vertex al-gebras associated to these automorphisms.The construc-tions in Refs.[7]and [9]are our special cases,which will be given in Subsecs.5.2and 5.3for completeness.5.1Construction of Lie Algebra Lsl (n,C )[θ]Denote E ij to be n ×n matrix whose (i,j )entry is 1and the others are zero.Set αk =E 11−E kk ,clearly,{αj ,E kl |j =2,...,n ;k,l =1,...,n ,and k =l }is a basis of sl (n,C ).The commutators of them are givenas [E ij ,E kl ]=δjk E il −δli E kjand[αj ,E kl ]=(δ1k E 1l −δ1l E k 1)−(δjk E jl −δjl E kj ).Fix a positive integral p and denote ωby exp(2π√−1/p ).Define an automorphism on the vector space sl (n,C )asθ1:sl (n,C )−→sl (n,C ),E kl −→ω(kl )E kl ,αs −→αs ,k,l =1,...,n ,s =2,...,n ,where (kl )is an integral and satisfies the formulas (ij )+(jk )=(ik ),(ij )=−(ji ),1≤i,j,k ≤n.Clearly,given (1j )any integer with j =2,...,n ,we can get a solution of the above formula by (ij )=(i 1)+(1j )=(1j )−(1i ).Note that δjk ω(il )=δjk ω(ij )+(jl )=δjk ω(ij )+(kl ),we can getθ1([E ij ,E kl ])=θ1(δjk E il −δli E kj )=δjk ω(il )E il −δli ω(kj )E kj=ω(ij )+(kl )(δjk E il −δli E kj )=ω(ij )+(kl )[E ij ,E kl ]=[θ1(E ij ),θ1(E kl )],No.1P-Twisted Affine Lie Algebras and Their Associated Vertex Algebras11 andθ1([αj,E kl])=θ1[(δ1k E1l−δ1l E k1)−(δjk E jl−δjl E kj)]= δ1kω(1l)E1l−δ1lω(k1)E k1− δjkω(jl)E jl−δjlω(kj)E kj= δ1kω(1l)+(k1)E1l−δ1lω(k1)+(1l)E k1− δjkω(jl)+(kj)E jl−δjlω(kj)+(jl)E kj=[(δ1k E1l−δ1l E k1)−(δjk E jl−δjl E kj)]ω(kl)=[θ1(αj),θ1(E kl)].Henceθ1is a Lie algebra automorphism.Andθp1=Id andθ1is invariant on the Carton subalgebraη=span C{αj|j=2,...,n}.Based on the automorphism,we get a gradationof sl(n,C)assl(n,C)=p−1j=0sl(n,C)(j),where sl(n,C)(j)={x∈sl(n,C)|θ1(x)=ωj x}.Consider C (t1/p) as a commutative Lie algebra and define a Lie algebra isomorphism asθ2:C (t1/p) −→C (t1/p) ,t j/p−→ω−j t j/p,j∈Z.Based on the isomorphism an extended p−automor-phism on Lsl(n,C)θ:Lsl(n,C)−→Lsl(n,C),A⊗f(t)−→θ1(A)⊗θ2(f(t)),where Lsl(n,C)=sl(n,C)⊗C((t1/p))is a formal loop algebra of sl(n,C).Now we can get an invariant subspace given by Lsl(n,C)[θ],which is a Lie algebra with the basis αj⊗t m,E kl⊗t(kl/p)+m with j=2,...,n;k,l=1,...,n; k=l;m∈Z.As above sections,we can construct a Z+-graded ver-tex operator algebra(V k(sl(n,C)[θ]),|0 ,T,Y(.,z)).5.2A Vertex Algebra of Affine Lie Algebrasl(2,C)[θ]Let g denote sl(2,C)with the basis{α2,E12,E21}, whereα2= 100−1 ,E12= 0100 ,E21= 0010 . The Lie brackets are given as[α2,E ij]=2(j−i)E ij,[E12,E21]=α2, where i=1,j=2,or i=2,j=1,and the others are zero. Letω=exp(2π√−1/3),and define an automorphism θ1:g−→g,α2−→α2,E12−→ωE12,E21−→ω−1E21.Since[θ1(α2),θ1(E ij)]=[α2,ωj−i E ij]=2ωj−i(j−i)E ij= 2(j−i)θ1(E ij)=θ1([α2,E ij])and[θ1(E12),θ1(E21)]=[ωE12,ω−1E21]=[E12,E21]=α2=θ1(α2)=θ1([E12,E21]),thenθ1is a Lie algebraic isomorphism. Clearlyθ31=Id,andθ1is invariant on Carton subalgebra η=span C{α2}. Considering C (t1/3) as a commutative Lie algebra, we define a Lie algebra isomorphismθ2:C (t1/3) −→C (t1/3) ,t j/3−→ω−j t j/3,j∈Z. Now we define3-automorphismθon L g=g⊗C (t1/3) based on the mapsθ1andθ2asθ:L g−→L g,A⊗f(t)−→θ1(A)⊗θ2(f(t)),theθ-invariant subspace is defined byL g[θ]=span C{α2⊗t n,E12⊗t n+(1/3),E21⊗t n+(2/3)|n∈Z}.Now we need to cheek that it is a Lie algebra.Since α2⊗t n,E12⊗t m+(1/3) =2E12⊗t n+m+(1/3), α2⊗t n,E21⊗t m+(2/3) =−2E21⊗t n+m+(2/3), E12⊗t n+(1/3),E21⊗t m+(2/3) =α2⊗t n+m+1, then the commutators of the basis are all in L g[θ].Hence, L g[θ]is a Lie ing the constructed method in above sections,we obtain a Z+-graded vertex algebra (V k(sl(2,C)[θ]),|0 ,T,Y(.,z)).5.3A Vertex Algebra of Affine Lie Algebrasl(3,C)[θ]Denote E ij to be3×3matrix whose(i,j)entry is1 and the others are zero.Setα1=E11−E22,x+α1=E12,x+α2=E23,x+α3=E13,α2=E22−E33,x−α1=E21,x−α2=E32,x−α3=E31.It can be checked that{αi,x±αj|i=1,2,j=1,2,3}is a basis of the Lie algebra sl(3,C)and its non-zero Lie brackets are[αi,αj]=0,i,j=1,2,[αi,x±αi]=±2x±αi,i=1,2,[αi,x±α3]=±x±α3,i=1,2,[αi,x±αj]=∓x±αj,i,j=1,2,i=j,[xαi,x−αi]=αi,i=1,2,[xα3,x−α3]=α1+α2,[x±α1,x±α2]=±x±α3,[x±α1,x∓α3]=∓x∓α2,[x±α2,x∓α3]=±x∓α1.Letω=exp(2π√−1/7).Define an automorphismθ1:sl(3,C)−→sl(3,C),αi−→αi,x±αj−→ω±.j x±αj, where i=1,2,and j=1,2,3.It is a Lie algebra isomorphism becauseθ1([αi,αj])=0=[θ1(αi),θ1(αj)],i,j=1,2,θ1([αi,x±αi ])=±2ω±i x±αi=[θ1(αi),θ1(x±αi)],i=1,2,θ1([αi,x±α3])=±ω±3x±α3=[θ1(αi),θ1(x±α3)],i=1,2,θ1 αi,x±αj =∓ω±j x±αj= θ1(αi),θ1 x±αj ,i,j=1,2,i=j,12DING Lu and ZHENG Zhu-Jun Vol.50θ1([xαi ,x−αi])=αi=[θ1(xαi),θ1(x−αi)],i=1,2,θ1([xα3,x−α3])=α1+α2=[θ1(xα3),θ1(x−α3)],θ1([x±α1,x±α2])=±ω±3x±α3=[θ1(x±α1),θ1(x±α2)],θ1 x±α1,x∓α3 =∓ω∓2x∓α2= θ1(x±α1),θ1 x∓α3 ,θ1 x±α2,x∓α3 =±ω∓1x∓α1= θ1(x±α2),θ1 x∓α3 .By calculations,we know thatθ71=Id andθ1is invariant on the Cartan subalgebraη=span C{αi|i=1,2}of sl(3,C). Define an automorphism of the commutative Lie algebra C (t1/7) asθ2:C (t1/7) −→C (t1/7) ,t j/7−→ω−j t j/7.Clearlyθ72=Id.On the Loop algebra Lsl(3,C)=sl(3,C)⊗C (t1/7) with the Lie bracket relations[A⊗f(t),B⊗g(t)]=[A,B]⊗(f(t)g(t)),we can define an automorphismθ:Lsl(3,C)−→Lsl(3,C),A⊗f(t)−→θ1(A)⊗θ2(f(t)),where A∈sl(3,C),f(t)∈C (t1/7) ,and its invariant subspaceLsl(3,C)[θ]=span C αi⊗t m,x+αj⊗t m+(j/7),x−αj⊗t m−(j/7)|i=1,2,j=1,2,3,m∈Z .Set −1 −j=−j/7and −1 +j=−1+(j/7),where j=1,2,3.For any m,n∈Z,αi⊗t m,αj⊗t n =0,i,j=1,2,αi⊗t m,x±αi⊗t n+ −1 ±i =±2x±αi⊗t m+n+ −1 ±i,i=1,2,αi⊗t m,x±α3⊗t n+ −1 ±3 =±x±α3⊗t m+n+ −1 ±3,i=1,2,αi⊗t m,x±αj⊗t n+ −1 ±j =∓x±αj⊗t m+n+ −1 ±j,i,j=1,2,i=j,x+αi⊗t m+ −1 +i,x−αi⊗t n+ −1 −i =αi⊗t m+n+ −1 +i+ −1 −i=αi⊗t m+n−1,i=1,2,x+α3⊗t m+ −1 +3,x−α3⊗t n+ −1 −3 =(α1+α2)⊗t m+n+ −1 +3+ −1 −3=(α1+α2)⊗t m+n−1,x±α1⊗t m+ −1 ±1,x±α2⊗t n+ −1 ±2 =±x±α3⊗t m+n+h±+ −1 ±3,x±α1⊗t m+ −1 ±1,x∓α3⊗t n+ −1 ∓3 =∓x∓α2⊗t m+n+h±+ −1 ∓2,x±α2⊗t m+ −1 ±2,x∓α3⊗t n+ −1 ∓3 =±x∓α1⊗t m+n+h±+ −1 ∓1,with h−=0,h+=−1,m,n∈Z.Therefore we get that Lsl(3,C)[θ]is a Lie algebra.Along the line of construction in the above sections,a Z+-graded vertex algebra V k sl(3,C)[θ] ,|0 ,T,Y(.,z) can be got on the vacuum representation space ofsl(3,C)[θ].AcknowledgmentsThe authors would like to thank Profs.Shi-Kun Wang and Xiao Zhang for their support and various kinds of help.References[1]R.E.Borcherds,A83(1986)3068.[2]R.E.Borcherds,Invent.Math.109(1992)405.[3]J.H.Conway and S.P.Norton,Bull.London Math.Soc.11(1979)308.[4]I.Frenkel,J.Lepowsky,and A.Meurman,Proc.Natl.A81(1984)3256.[5]I.Frenkel,J.Lepowsky,and A.Meurman,Vertex Oper-ator Algebras and the Monster,Pure and Appl.Math.Vol.134,Academic Press,Boston(1988).[6]A.Beilinson and V.Drinfeld,Chiral Algebra,Colloq.Publ.51AMS(2004).[7]Wen-Feng Dong,Yan-Jun Chu,and Zhu-Jun Zheng,Re-alization of Vertex Operators of7-Twisted Affine Lie Al-gebrasl(3,C)[θ],preprint.[8]Ke-Ping Lu,Yan-Jun Chu,Zhu-Jun Zheng,and Wen-Feng Dong,Sci.in China(2005)295.[9]Xia Li and Zhu-Jun Zheng,A vertex algebra of3-twistedaffine Lie algebrasl(2,C)[θ],preprint.[10]Edward Frenkel and David Ben-Zvi,Vertex Algebras andAlgebraic Curves,Math.Surv.Mon.Vol.88,AMS(2004).[11]Y.Z.Huang,Two-Dimensional Conformal Geometry andVertex Operator Algebras,Progress in Mathematics148, Birkhauser,Boston(1997).[12]I.Frenkel,J.Lepowsky,and A.Meurman,Vertex Oper-ator Calculus,in:Mathematical Aspects of String The-ory,Proc.1986Conference,San Diego,ed.by S.-T.Yau, World Scientific,Singapore(1987)pp.150–188.[13]V.Kac,Vertex Algebras for Beginners,2nd.ed.,AMS(1998).。