Three Schur functors related to pre-Lie algebras

TL;DR

This paper provides explicit combinatorial descriptions of three Schur functors related to pre-Lie algebras, offering new insights into their universal enveloping algebras, modules, and cohomology interpretations.

Contribution

It introduces novel combinatorial descriptions of Schur functors associated with pre-Lie algebras, enhancing understanding of their universal structures and cohomology.

Findings

Functorial description of the universal enveloping pre-Lie algebra

Functorial descriptions of the universal multiplicative enveloping algebra and Kähler differentials

Cohomology of pre-Lie algebras interpreted as a derived functor

Abstract

We give explicit combinatorial descriptions of three Schur functors arising in the theory of pre-Lie algebras. The first of them leads to a functorial description of the underlying vector space of the universal enveloping pre-Lie algebra of a given Lie algebra, strengthening the PBW theorem of Segal. The two other Schur functors provide functorial descriptions of the underlying vector spaces of the universal multiplicative enveloping algebra and of the module of K\"ahler differentials of a given pre-Lie algebra. An important consequence of such descriptions is an interpretation of the cohomology of a pre-Lie algebra with coefficients in a module as a derived functor for the category of modules over the universal multiplicative enveloping algebra.