Vertex Algebras and the Equivariant Lie Algebroid Cohomology

TL;DR

This paper develops a vertex-algebraic framework for Lie algebroid cohomology, generalizing classical concepts to a chiral setting and enabling new computations, especially for Poisson-Lie groups.

Contribution

It introduces VSA-inductive sheaves and constructs a vertex-algebraic analogue of the Lie algebroid complex, extending equivariant cohomology to a chiral context.

Findings

Constructed a vertex-algebraic analogue of the Lie algebroid complex.

Generalized equivariant Lie algebroid cohomology to a vertex-algebraic form.

Computed cohomology in special cases, including Poisson-Lie groups.

Abstract

A vertex-algebraic analogue of the Lie algebroid complex is constructed, which generalizes the "small" chiral de Rham complex on smooth manifolds. The notion of VSA-inductive sheaves is also introduced. This notion generalizes that of sheaves of vertex superalgebras. The complex mentioned above is constructed as a VSA-inductive sheaf. With this complex, the equivariant Lie algebroid cohomology is generalized to a vertex-algebraic analogue, which we call the chiral equivariant Lie algebroid cohomology. In fact, the notion of the equivariant Lie algebroid cohomology contains that of the equivariant Poisson cohomology. Thus the chiral equivariant Lie algebroid cohomology is also a vertex-algebraic generalization of the equivariant Poisson cohomology. A special kind of complex is introduced and its properties are studied in detail. With these properties, some isomorphisms of cohomologies are developed, which enables us to compute the chiral equivariant Lie algebroid cohomology in some cases. Poisson-Lie groups are considered as such a special case.