Duality for graded Lie algebras

TL;DR

This paper extends the classical Poincare duality for Lie algebra cohomology to graded and differential graded Lie algebras, revealing new duality structures and Calabi-Yau properties in their derived categories.

Contribution

It establishes a graded Lie algebra analogue of Poincare duality, computes specific cohomology with universal enveloping algebra coefficients, and links these results to Calabi-Yau structures in derived categories.

Findings

Cohomology of graded Lie algebras with coefficients in U(g) is one-dimensional.

Unimodular graded Lie algebras have Calabi-Yau derived categories.

Generalization of Poincare duality to rational infinity local systems on elliptic spaces.

Abstract

A well-known and old result of Hazewinkel and Koszul states that the cohomology of a finite-dimensional Lie algebra is isomorphic, up to a suitable shift, to its twisted homology, a Lie-theoretical version of Poincare duality. This paper establishes an analogue of this result for graded (or super) Lie algebras and, more generally, differential graded Lie algebras. Closely related to this result is a calculation of the cohomology of a graded Lie algebra g with coefficients in its universal enveloping algebra U(g) as a one-sided module. This cohomology turns out to be one-dimensional and serves as a dualizing module for the cohomology of g. Moreover, it is shown that for a unimodular graded Lie algebra g, the derived category of U(g) has a Calabi-Yau structure. As a consequence, the category of rational infinity local systems on a simply connected topological space with totally finite-dimensional rational homotopy groups, has a Calabi-Yau structure, a generalization of Poincare duality for elliptic spaces.