Lie algebra homology with coefficients tensor products of the adjoint representation in relative polynomial degree 2

TL;DR

This paper computes the second relative polynomial degree homology of free Lie algebras with tensor product coefficients of the adjoint representation, revealing new structural features and confirming a conjecture.

Contribution

It provides the first explicit calculation of homology in relative degree 2, advancing understanding of polynomial outer functors and homological properties of free Lie algebras.

Findings

Homology in relative degree 2 has interesting structural features.

Confirms a conjecture of Gadish and Hainaut.

Supports the polynomial degree splitting in Lie algebra homology.

Abstract

The homology of free Lie algebras with coefficients in tensor products of the adjoint representation working over Q contains important information on the homological properties of polynomial outer functors on free groups. The latter category was introduced in joint work with Vespa, motivated by the study of higher Hochschild homology of wedges of circles. There is a splitting of this homology by polynomial degree (for polynomiality with respect to the generators of the free Lie algebra) and one can consider the polynomial degree relative to the number of tensor factors in the coefficients. It suffices to consider the Lie algebra homology in homological degree one; this vanishes in relative degree 0 and is readily calculated in relative degree 1. This paper calculates the homology in relative degree 2, which presents interesting features. This confirms a conjecture of Gadish and Hainaut.