论文标题
谎言群体上的单数连锁店和卡坦关系II
Singular chains on Lie groups and the Cartan relations II
论文作者
论文摘要
让$ g $是一个简单地连接的谎言组，lie代数$ \ mathfrak {g} $，用$ \ mathrm {c} _ {\ bulter}（g）$ dg hopf代数为$ g $。在同伴论文中表明，在$ \ mathrm {c} _ {\ bullet}（g）上，足够光滑的模块的类别等于$ \ mathbb {t} \ mathfrak {g Mathfrak {g} $，dg lie algebra的$ \ mathbb {t} \ mathfrak {t} \ mathfrak {t} \ mathfrak {t} \ mathbB {在本文中，我们表明，如果$ g $是紧凑的，那么类别的等效性可以扩展到$ \ mathsf {a} _ {\ infty} $ - 相应的DG类别的准等值。作为中级步骤，我们构建了一个$ \ Mathsf {a} _ {\ infty} $ - Quasi-Isomorphism of $ g $与$ g $相关的Bott-Shulman-Stasheff DG Algebra与$ \ Mathrm {C} c} c} c} c} in $ g $的DG代数相关的dg代数。证明中的主要成分是van est Map和gugenheim的$ \ mathsf {a} _ {\ infty} $版本的de rham定理的版本。
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.
