Koszul Duality for modules over Lie algebra

TL;DR

This paper establishes a Koszul duality relationship between the invariants and equivariant cohomology of modules over a reductive Lie algebra, expanding the algebraic understanding of these structures.

Contribution

It introduces a new duality framework connecting invariants and equivariant cohomology for modules over Lie algebras, generalizing previous results.

Findings

Proves Koszul duality between invariant and equivariant cohomology.

Defines cohomology of invariants and equivariant cohomology in this context.

Establishes the duality under specific algebraic conditions.

Abstract

Let $g$ be a reductive Lie algebra over a field of characteristic zero. Suppose $g$ acts on a complex of vector spaces $M$ by $i_\lambda$ and $L_\lambda$, which satisfy the identities as contraction and Lie derivative do for smooth differential forms. Out of this data one defines cohomology of the invariants and equivariant cohomology of $M$. We establish Koszul duality between each other.