Singular chains on Lie groups and the Cartan relations I

Abstract

Let $G$ be a simply connected Lie group with Lie algebra $\\mathfrak{g}$ and denote by $\\mathrm{C}_{\\bullet}(G)$ the DG Hopf algebra of smooth singular chains on $G$. In a companion paper it was shown that the category of sufficiently smooth modules over $\\mathrm{C}_{\\bullet}(G)$ is equivalent to the category of representations of $\\mathbb{T} \\mathfrak{g}$, the DG Lie algebra which is universal for the Cartan relations. In this paper we show that, if $G$ is compact, this equivalence of categories can be extended to an $\\mathsf{A}_{\\infty}$-quasi-equivalence of the corresponding DG categories. As an intermediate step we construct an $\\mathsf{A}_{\\infty}$-quasi-isomorphism between the Bott-Shulman-Stasheff DG algebra associated to $G$ and the DG algebra of Hochschild cochains on $\\mathrm{C}_{\\bullet}(G)$. The main ingredients in the proof are the Van Est map and Gugenheim s $\\mathsf{A}_{\\infty}$ version of De Rham s theorem.