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

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.

💡 Research Summary

The paper investigates the Lie algebra homology of a free Lie algebra (Lie(V)) with coefficients in tensor powers of its adjoint representation, i.e. (H_{*}(Lie(V);,Lie(V)^{\otimes r})) over the rational numbers. Because the homology of a free Lie algebra vanishes in degrees higher than one, the problem reduces to understanding the degree‑one homology (H_{1}). This homology is naturally a functor of the underlying vector space (V) and carries an action of the symmetric group (S_{r}) permuting the tensor factors.

The authors first recall the standard splitting of Lie algebra homology by polynomial degree: for each homogeneous component (