Weyl Modules for Toroidal Lie Algebras

Abstract

In this paper, we extend the notion of Weyl modules for twisted toroidal Lie algebra T (µ). We prove that the level one global Weyl modules of T (µ) are isomorphic to the tensor product of the level one representation of twisted affine Lie algebras and certain lattice vertex algebras. As a byproduct, we calculate the graded character of the level one local Weyl modules of T (µ).Notations:• The sets of complex numbers, integers, non-negative integers, and positive integers are denoted by C, Z, Z ≥0 , and Z >0 , respectively.and the sets Z n , Z n ≥0 , and Z n >0 are defined similarly. • For n ≥ 2, we denote the elements (m 1 , . . . , m n ) ∈ Z n and (m 2 , . . . , m n ) ∈ Z n−1 by m and m, respectively. • Let A n := C[t ±1 1 , t ±1 2 , . . . , t ±1 n ] be the set of Laurent polynomial ring in n variables. • For m = (m 1 , . . . , m n ) ∈ Z n and m = (m 2 , . . . , m n ) ∈ Z n−1 the notation t m and t m represents t m 1 1 • • • t mn n and t m 2 2 • • • t mn n , respectively, in A n . • Let U(g) denotes the universal enveloping algebra of g and L n (g) denotes the Lie algebra g ⊗ A n with commutator given by [x ⊗ t m , y ⊗ t n ] = [x, y] ⊗ t m+n .