On fundamental structure underlying Lie algebra homology with coefficients tensor products of the adjoint representation

TL;DR

This paper reveals a fundamental algebraic structure in Lie algebra homology with tensor product coefficients, especially for free Lie algebras, connecting it to DG categories, bimodules, and outer functors over free groups.

Contribution

It constructs a DG category from the Lie operad's PROP and links Lie algebra homology to outer functors, providing a new algebraic perspective.

Findings

Homology is related to a two-term complex of bimodules.

The degree one homology corresponds to an outer functor.

Provides an algebraic explanation contrasting previous approaches.

Abstract

This paper exhibits fundamental structure underlying Lie algebra homology with coefficients in tensor products of the adjoint representation, mostly focusing upon the case of free Lie algebras. The main result yields a DG category that is constructed from the PROP associated to the Lie operad. Underlying this is a two-term complex of bimodules over this PROP; it is a quotient of the universal Chevalley-Eilenberg complex. The homology of this DG category is intimately related to outer functors over free groups (introduced in earlier joint work with Vespa). This uses the author's previous results relating functors on free groups to representations of the PROP associated to the Lie operad. This gives a direct algebraic explanation as to why the degree one homology should correspond to an outer functor. Hitherto, the only known argument relied upon the relationship with the higher Hochschild homology functors that arise from the work of Turchin and Willwacher.