A category of modules for the full toroidal Lie algebra. (English) Zbl 1201.17016

Billig, Yuly

Int. Math. Res. Not. 2006, No. 23, Article ID 68395, 46 p. (2006).

Toroidal Lie algebras are natural generalizations of affine Kac-Moody algebras. Not only they are very interesting algebraic structures, but also proving themselves to be useful in mathematics and physics. For the full toroidal Lie algebra \(g\), the author defined a category of bounded \(g\)-modules with finite dimensional weight spaces, and proved that the irreducible modules in this category are characterized by its top and classified them accordingly. Furthermore, the author established that every irreducible module in this category admits a simple VOA module structure for the vertex operator algebra \(V(T_0)\), a VOA as a product of two related VOA’s as defined in the paper, here \(T_0\) is the ring consisting of multi-variable Laurent polynomials. The results presented in the paper are very helpful to understand the structures and their representation theories of toroidal Lie algebras.

Reviewer: Xiandong Wang (Qingdao)

MSC:

17B69	Vertex operators; vertex operator algebras and related structures
17B67	Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras

Keywords:

toroidal Lie algebras; vertex algebras; affine Lie algebras; irreducible representation