Gradation Shifting Toroidal Lie Algebras And Representations For Baby-TKK Algebra

The extended affine Lie algebras (EALAs for short) are important graded Liealgebras,which include finite dimensional simple Lie algebras,affine Kac-MoodyLie algebras,Lie algebras coordinated by Laurent polynomial torus and quantumtorus,and a class of Tits-Kantor-Kocher algebras coordinated by Jordan torus.It iswell known that toroidal Lie algebra (with certain central elements and derivationsadded) is a generalization of non-twist affine Kac-Moody algebra.It is an EALAcoordinated by Laurent polynomial torus A_v=C[t₁^±1,...,t_v^±1].The classification ofirreducible integrable represetations for affine Kac-Moody algebras and toroidal Liealgebras have been studied by many researchers,such as [C,CP1,E1,E2,E3,E4,EJ]etc.The irreducible integrable represetations include a class of represetationswhere central elements act as zero.These represetations are related to the finitedimensional irreducible represetations for the loop algebra or multi-loop algebra.Thus it is important to consider the finite dimensional irreducible represetations forsome infinite dimensional Lie algebras.In this paper,we define a class of infinite dimensional Lie algebras,whichgeneralize the toroidal Lie algebras of type B_l and D_l.Let so(n,C) for n≥3 be thecomplex orthogonal Lie algebra with basis{α_ij:=e_ij-e_ji|1≤i≠j≤n}where e_ij is the matrix with unit 1 in (i,j)-entry and 0 elsewhere.Let A be anycommutative associative algebra over C with identity,and E₁,...,E_n∈A be anyfixed elements.We define a Lie algebra on so(n,C) (?) A with the following bilinearmultiplication:[α_ij (?) f,α_jk(?)g]=α_ik(?) E_jfg,[α_ij(?),f,α_ij(?) g]= [α_ij (?) f,α_kl(?) g]=0,where i,j,k,l distinct,and f,g∈A.We call it gradation shifting toroidal Lie al-gebra,and denote it by L_n(E₁,...,E_n).Note that if A = A_v,E₁=…=E_n = 1,L_n(1,...,1) is a multi-loop algebra,and the universal central extension is a toroidalLie algebra.The case n = 3 is given in [LT].In [LT],the authors give the derivationsand universal central extension of L₃(t^s₁,t^s₂,1),and give a class of vertex operator represetations for L₃(t₁,t₂,1) with two varibles.And in [CLT],the authors obtainthe automorphism group and a class of Wakimoto represetations for L(t₁,t₂,1).Inparticular,when n = 3,the authors of [SG]generalize the difinition to noncom-mutative associative algebra A.The gradation shifting toroidal Lie algebra is ageneralization of [LT].We prove that gradation shifting toroidal Lie algebra L_n(E₁,...,E_n) is perfectif and only if∑_k≠i,jE_kA = A,for any i＜j.In particular,if the elements E₁,...,E_nare units of A,then there exist s₁,...,s_n∈Z₂^v,such thatL_n(E₁,...,E_n)≌L_n(t^s₁,...,t^s_n).We only consider the case E_i's are units.In this case,L_n(t^s₁,...,t^s₂) is a finitelygenerated and ((Z/2Z)ⁿ,1/2 Z^v)-graded perfect Lie algebra.We obtain the derivationsand universal central extension of gradation shifting toroidal Lie algebra.Becausethe case n = 4 is very different from the other cases,we divide our main result intotwo cases.In what follows,we discuss the finite dimensional irreducible represetationsfor the gradation shifting toroidal Lie algebra.We define a Lie algebra surjectivehomomorphism from gradation shifting toroidal Lie algebra L_n(t^s₁,...,t^s_n) ontosemisimple Lie algebra so(n,C)^⊕N (direct sum of N copies of so(n,C)).This im-plies that any finite dimensional irreducible represetation for so(n,C)^⊕N can belifted to finite dimensional irreducible represetation for gradation shifting toroidalLie algebra L_n(t^s₁,...,t^s_n) through this surjective homomorphism.Conversely,wealso prove that any finite dimensional irreducible represetation for L_n(t^s₁,...,t^s_n)comes from the irreducible represetation for so(n,C)^⊕N.We note that,there doesnot exist usual Cartan subalgebra in gradation shifting toroidal Lie algebra.Theproof is different from the case of multi-loop algebras considered by S.Eswara Rao.So instead of choosing a Cartan subalgebra,we choose another finite dimensionalabelian subalgebra.By citeAABGP,we know that the extended affine root system of type A₁ isdecided by the semi-lattice A in the Euclidean space.If nullity is 0,the EALAmust be finite dimensional simple Lie algebra,and if nullity is 0,the EALA mustbe affine Kac-Moody algebra.Therefore,we first consider the theories for the casenullity= 2.Note that in the Euclidean space R²,there are only two non-similar semi-lattices:the lattice Z² and the smallest possible (non-lattice) semi-lattice S.The semi-lattice S corresponds to the baby Tits-Kantor-Koecher algebra (?)(J(S)).Thebaby Tits-Kantor-Koecher algebra (with certain central elements and derivationsadded) is related to the"smallest"extended affine Lie algebras other than the finitedimensional simple Lie algebras and the affine Kac-Moody algebras.In [T3]and[MT1],vertex operator represetations for the universal central extension of the TKKalgebra (?)(J(S)) are obtained,and a class of Wakimoto represetations for (?)(J(S))are given in [MT2].For the Lie algebra sl_l+1(C_Q) coordinated by a quantum torus C_Q,whereQ = (q_ij)_{1≤i,j≤2},(q_ij = q^i-j,q is Nth-primitive root of unity),the classificationof the finite dimensional irreducible represetations are given in [EB].In [EB],theauthors prove that any finite dimensional irreducible represetation for sl_l+1(C_Q)comes from finite dimensional irreducible represetation for the semi-simple Lie alge-bra⊕sl_N(l+1) (C) (a suitable number of copies) through certain surjective Lie algebrahomomorphism.We note that when q = -1,TKK algebra (?)(J(Z²)) is isomorphicto sl₂(C_Q).Therefore,from [EB],we know that any finite dimensional irreduciblerepresetation for the TKK algebra (?)(J(Z²)) comes from finite dimensional irre-ducible represetation for the semi-simple Lie algebra⊕sl₄(C) (a suitable number ofcopies).We generalize this result to baby-TKK algebra (?)(J(S)).We prove that anyfinite dimensional irreducible represetation for baby-TKK algebra (?)(J(S)) comesfrom finite dimensional irreducible represetation for the semi-simple Lie algebra⊕sp₄(C) (a suitable number of copies).