Algebra of q-difference operators, affine vertex algebras, and their modules

作者：Guo, Hongyan^*

来源：Journal of Algebra, 2021, 585: 69-88.

DOI：10.1016/j.jalgebra.2021.06.007

• q-analog Virasoro-like algebra
• Generalized affine Lie algebras
• Vertex algebras
• Modules

In this paper, we explore a canonical connection between the algebra of q-difference operators (V) over tilde (q), affine Lie algebras and affine vertex algebras associated to certain subalgebra A of the Lie algebra gl(infinity). We also introduce and study a category R of (V) over tilde (q)-modules. More precisely, we obtain a realization of (V) over tilde (q) as a covariant algebra of the affine Lie algebra (A) over cap*, where A* is a 1-dimensional central extension of A. We prove that restricted (V) over tilde (q)-modules of level l(12) correspond to Z-equivariant f-coordinated quasi-modules for the vertex algebra V-(A) over tilde (l(12), 0), where (A) over tilde is a generalized affine Lie algebra of A. In the end, we show that objects in the category Rare restricted (V) over tilde (q)-modules, and we classify simple modules in the category R.