Singular chains on Lie groups and the Cartan relations I

Singular chains on Lie groups and the Cartan relations II

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. A…

Maurer-Cartan elements in the Lie models of finite simplicial complexes

In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex…

A Lie theoretic approach to the twisting procedure and Maurer-Cartan simplicial sets over arbitrary rings

The Deligne-Getzler-Hinich--$\infty$-groupoid or Maurer-Cartan simplicial set of an $L_\infty$-algebra plays an important role in deformation theory and many other areas of mathematics. Unfortunately, this construction only works over a field of characteristic $0$. The goal of this paper is to show that the notions of Maurer-Cartan equation and Maurer-Cartan simplicial set can be defined for a much larger number of operads than just the $L_\infty$-operad. More precisely, we show that the Koszul dual of every unital Hopf cooperad (a cooperad in the category of unital associative algebras) with …

The infinity Quillen functor, Maurer-Cartan elements and DGL realizations

We show an alternative construction of the cosimplicial free complete diferential graded Lie algebra $\mathfrak{L}_\bullet=\widehat{\mathbb{L}}(s^{-1}Δ^\bullet)$ based on a new Lie bracket formulae for Lie polynomials on a general tensor algebra. Based on it,we prove that for any complete differential graded Lie algebra $L$, its geometrical realization $\langle L\rangle=\text{Hom}_{\text{cdgl}}(\mathfrak{L}_\bullet,L)$ is isomorphic to its nerve $γ_\bullet(L)$, a deformation retract of the Getzler-Hinich realization $\text{MC}(\mathscr{A}_\bullet\widehat{\otimes} L)$.

关于指数群与Maurer-Cartan空间

On exponential groups and Maurer-Cartan spaces

本文给出指数群与Maurer-Cartan空间基本性质的简明阐述，证明Maurer-Cartan空间是指数群的去环化，并由此证明该函子是Quillen函子的同伦逆。

Equivariant characteristic forms in the Cartan model and Borel equivariant cohomology

We show the compatibility of the differential geometric and the topological construction of equivariant characteristic classes for compact Lie groups. Our analysis motivates a differential geometric construction for equivariant characteristic classes in the non-compact case. This compatibility is generally assumed and used in various cases, but there is only a proof for compact connected Lie groups in the literature, see the work of Raoul Bott and Loring Tu. Our proof applies and generalizes ideas of Johan Dupont and Ezra Getzler.