Singular chains on Lie groups and the Cartan relations I

TL;DR

This paper establishes a natural equivalence between modules over the DG-algebra of singular chains on a simply connected Lie group and representations of a related DG-Lie algebra, extending classical Lie theory.

Contribution

It introduces a new categorical equivalence linking smooth modules over singular chains with DG-Lie algebra representations, generalizing the classical Lie group-Lie algebra correspondence.

Findings

Categories of modules and representations are equivalent.

Extension to an A-infinity equivalence in the compact case.

Generalizes classical Lie theory to DG-algebra and DG-Lie algebra contexts.

Abstract

Let $G$ be a simply connected Lie group with Lie algebra $\mathfrak{g}$. We show that the following categories are naturally equivalent. The category $\mathsf{Mod}(C(G))$, of sufficiently smooth modules over the DG-algebra of singular chains on $G$. The category $\mathsf{Rep}(Tg)$ of representations of the DG-Lie algebra $T\mathfrak{g}$, which is universal for the Cartan relations. This equivalence extends the correspondence between representations of $G$ and representations of $\mathfrak{g}$. In a companion paper we show that in the compact case, the equivalence can be extended to an $\mathsf{A}_\infty$ equivalence of DG-categories.