Cyclic cohomology of Lie algebras

TL;DR

This paper explores the structure of stable-Yetter-Drinfeld modules over Lie algebras, establishing isomorphisms with modules over their enveloping algebras and connecting various complexes in Lie algebra cohomology.

Contribution

It defines stable-Yetter-Drinfeld modules over Lie algebras, proves their category is isomorphic to that over enveloping algebras, and introduces a mixed complex linking several known complexes.

Findings

Established isomorphism between categories of modules

Defined a new mixed complex for Lie algebras

Connected known complexes like Weil and Koszul to this framework

Abstract

In this paper we aim to understand the category of stable-Yetter-Drinfeld modules over enveloping algebra of Lie algebras. To do so, we need to define such modules over Lie algebras. These two categories are shown to be isomorphic. A mixed complex is defined for a given Lie algebra and a stable-Yetter-Drinfeld module over it. This complex is quasi-isomorphic to the Hopf cyclic complex of the enveloping algebra of the Lie algebra with coefficients in the corresponding module. It is shown that the (truncated) Weil algebra, the Weil algebra with generalized coefficients defined by Alekseev-Meinrenken, and the perturbed Koszul complex introduced by Kumar-Vergne are examples of such a mixed complex.