Representations Of The Half Lattice Vertex Algebra And Graded Automorphism Group Of TKK Algebra

Vertex algebras are a new class of mathematical object developed in the end of the twentieth century. Their definition is well motivated both by representationtheory of affine Kac-Moody algebras and the conformal field theory in physics, (cf. [Bol, MS])The lattice vertex algebras form one of the most important and fundamentclasses of vertex algebras. S. Berman, C. Dong and S. Tan studied the representation theory for certain " half lattice " vertex algebras which are related to the studying of the representation theory for toroidal Lie algebras([BDT]). Let L be an even lattice and let V_L be the associated lattice vertex algebra. As a vector space, V_L is the tensor product of a symmetric algebra S(?) with a group algebra C[L] where H = (?). The lattice L considered in [BDT] is spanned by c_i, d_i for i = 1,…,Ï…with the Z-bilinear form determined by (c_i,c_j) = (d_i,d_j) = 0 and (c_i,d_j) =δ_i,j. The half lattice vertex algebra V is then defined to bewhere L_C = (?). The half lattice vertex algebra V is a vertex subalgebraof the lattice vertex algebra V_L.The authors of [BDT] defined an associative algebra A, which is generated by e_αand d_i, subject to the relationsforα,β∈L_C, 1≤i,j≤ν. Furthermore, they gave a one-to-one correspondence between the (irreducible) .A-modules and the (irreducible) modules of the half lattice vertex algebra V. That is, they could construct an (irreducible) V-module from an (irreducible) ,A-module and construct an (irreducible) A-module from an (irreducible) V-module. This also means that the work to construct more representations of the associative algebra A is meaningful. In the first chapter of this paper, we first define an associative algebra A_Q. Let Q = (q_ij) be anyυ×υmatrix of nonzero complex numbers which satisfyand A_Q be the associative algebra generated by e_α,d_i, subject to the relationsforα=(?),β=(?),1≤i,j≤υ. When all entries in Q are 1, the associative algebra A_Q is nothing but the associative algebra A. Moreover, We construct two classes of irreduible A_Q-modules: V(α₁,…,α_Ï…-1,b) and V(ā). We also study their automorphism groups.The classification of extended affine Lie algebras of type A₁ depends on the TKK algebras constructed from semilattices of Euclidean spaces. One can define a Jordan algebra J(S) from a semilattice S of R^Ï…(υ≥1), and then construct an extended affine Lie algebra of type A₁ from the TKK algebra (?)(J(S)) which is obtained from the Jordan algebra J(S) by the so-called Tits-Kantor-Koecher construction. The authors of [AABGP] have proved that there is a one to one correspondence between the set of all similarity classes of semilattices in R^Ï…and the set of all isomorphism classes of extended affine root systems of type A₁ with nullityÏ…. In Euclidean space R² there are only two non-similar semilattices S and S', where S is a lattice and S' is a non-lattice semilattice. The Jordan algebras J(S) and J(S') both have a natural Z²-gradation. These gradations naturally induce Z²-gradations on TKK algebra (?)(J(S)) and Baby TKK algebra (?)(J(S')). In the second chapter of this paper, we study the Z²-graded automorphism groups of (?)(J(S)) and (?)(J(S')) respectively.