Homotopy Lie algebras and coherent infinitesimal 2-braidings

TL;DR

This paper explores the structure of homotopy Lie algebras, constructing symmetric monoidal dg-categories of modules, and demonstrates the coherence of infinitesimal 2-braidings derived from 2-shifted Poisson structures.

Contribution

It explicitly constructs the dg-category of modules over homotopy Lie algebras and establishes a dg-equivalence with semi-free dg-modules over the Chevalley-Eilenberg algebra, including coherence results.

Findings

Construction of symmetric monoidal dg-category of modules

Explicit differential of Chevalley-Eilenberg algebra

Equivalence between module categories

Abstract

Given a homotopy Lie algebra (i.e. an $L_\infty$-algebra) $\mathfrak{g}$, we show concretely how the Lada-Markl $\mathfrak{g}$-modules (i.e. representations) assemble into a symmetric monoidal dg-category. Considering the homotopy 2-category of that dg-category, we construct infinitesimal 2-braidings from 2-shifted Poisson structures then show that such infinitesimal 2-braidings are coherent in Cirio and Faria Martins' sense. We then explicitly determine the differential of the Chevalley-Eilenberg algebra associated with a finite-dimensional homotopy Lie algebra and construct the symmetric monoidal dg-equivalence between the category of representations and the category of semi-free dg-modules over the Chevalley-Eilenberg algebra.