Koszul duality for modules over Lie algebras

Abstract

Let $\mathfrak {g}$ be a reductive Lie algebra over a field of characteristic zero. Suppose that $\mathfrak {g}$ acts on a complex of vector spaces $M\sp \bullet$ by $i\sb \lambda$ and $\mathscr {L}\sb \lambda$, which satisfy the same identities that contraction and Lie derivative do for differential forms. Out of this data one defines the cohomology of the invariants and the equivariant cohomology of $M\sp \bullet$. We establish Koszul duality between them.