Homotopy Lie algebras and coherent infinitesimal 2-braidings
Cameron Kemp
TL;DR
This work develops a higher-categorical representation theory for $L_\infty$-algebras by constructing a symmetric monoidal dg-category of representations $\mathsf{Rep}(\mathfrak{g})$ from a homotopy Lie algebra $\mathfrak{g}$ and endowing it with a coherent infinitesimal $2$-braiding derived from a $2$-shifted Poisson structure. It introduces a detailed intertwiners framework and the monoidal structure on representations, then shows how a $2$-shifted Poisson structure induces a totally $\gamma$-equivariant coherent inf2braiding on the homotopy 2-category. In the finite-dimensional setting, it identifies the Chevalley–Eilenberg algebra $\mathrm{CE}_{\mathfrak{g}}$ as a completed semi-free CDGA and proves a symmetric monoidal dg-equivalence between $\mathsf{Rep}(\mathfrak{g})$ and the category of completed semi-free dg-modules over $\mathrm{CE}_{\mathfrak{g}}$, tying representations to derived geometry. This framework advances the program of higher quantum groups and deformation quantization of monoidal structures in a homotopical, derived setting.
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.