Vertex operator algebras associated to modified regular representations of affine Lie algebras

Details Vertex operator algebras associated to modified regular representations of affine Lie algebras Vertex operator algebras associated to modified regular representations of affine Lie algebras

2006-11-17

arxiv.org/abs/math/0611517v2

Mathematics

Quantum Algebra

29 pages

Scientific paper

Let $G$ be a simple complex Lie group with Lie algebra $\mf g$ and let $\af$ be the affine Lie algebra. We use intertwining operators and Knizhnik-Zamolodchikov equations to construct a family of $\N$-graded vertex operator algebras associated to $\mf g$. They are $\af \oplus \af$-modules of dual levels $k, \bar k \notin \Q$ in the sense that $k + \bar k = -2 h^\vee$ where $h^\vee$ is the dual Coxeter number of $\mf g$. Its conformal weight 0 component is the algebra of regular functions on $G$. This family of vertex operator algebras were previously studied by Arkhipov-Gaitsgory and Gorbounov-Malikov-Schechtman from different points of view. We show that the vertex envelope of the vertex algebroid associated to $G$ and level $k$ is isomorphic to the vertex operator algebra we constructed above when $k$ is irrational. The case of integral central charges is also discussed.

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