Vertex operator algebras arising from affine Lie algebras.

摘要

We study a family of vertex operator algebras associated to modified regular representations of affine Lie algebras. These vertex operator algebras admit two copies of the affine Lie algebras acting on them with dual central charges and the top levels coincide with the regular functions on the Lie groups.;In the generic case, we construct the vertex operators based on the properties of intertwining operators and Knizhnik-Zamolodchikov equations. We also show that the enveloping algebra of the vertex algebroid associated to the Lie group and a fixed level is isomorphic to the vertex operator algebra we constructed using intertwining operators.;The equivalence of categories behind the VOA structure suggests that there is a strong connection between the module structure of these vertex operator algebras in rational levels and the structure of the regular representations of the "big" quantum groups at roots of unity. In fact, we show that the quantum function algebra admits an increasing filtration with factors isomorphic to the tensor products of the dual of the Weyl modules. Then we use the results on quantum function algebras and the standard semi-regular module to prove the existence of canonical filtrations of this family of vertex operator algebras in rational levels.