Let G be a simply connected Lie group with Lie algebra \({\mathfrak {g}}\). We show that the following categories are naturally equivalent. The category \(\mathbf{Mod }({\text {C}}_{{\bullet }}(G))\) of sufficiently smooth modules over the DG algebra of singular chains on G. The category \(\mathbf{Rep }(\mathbb {T}{\mathfrak {g}})\) of representations of the DG Lie algebra \(\mathbb {T}{\mathfrak {g}}\), 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 will show that in the compact case, the equivalence can be extended to a quasi-equivalence of DG categories.

设G是一个与李代数\({\mathfrak {g}}\)的单连通李群。我们证明以下类别自然是等价的。类别\（\ mathbf {国防部}（{\文本{C}} _ {{\子弹}}（G））\）的充分平滑过度奇异链的DG代数模块ģ。DG李代数\(\mathbb {T}{\mathfrak {g}}\)表示的范畴\(\mathbf{Rep }(\mathbb {T}{\mathfrak {g}}) \)，其中对于嘉当关系是普遍的。这种等价扩展了G 的表示与\({\mathfrak {g}}\) 的表示之间的对应关系. 在配套论文中，我们将展示在紧凑情况下，等价可以扩展到 DG 类别的准等价。