Singular chains on Lie groups and the Cartan relations I

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.

Changes in notation. Added references. All comments are welcome