Singular Chains on Lie Groups and the Cartan Relations

Explore the intricate connections between Lie groups, singular chains, and Cartan relations in this mathematics seminar. Delve into the description of representations of C(G) using the differential graded Lie algebra Tg, and uncover its relationship to Chern-Weil theory and higher local systems on the classifying space BG. Learn about the group ring, singular chains, and an equivalence of categories. Examine the proof's key components, including Chen's iterated integrals, Bott-Shulman-Stasheff algebra, and the non-commutative Weil algebra. Investigate practical examples such as the trivial representation and the free loop space of BG to solidify understanding of these advanced mathematical concepts.

Intro
Outline
Lie theory
The group ring and singular chains
The Cartan relations
An equivalence of categories
How the proof works
Chern-Weil theory
Chern-Weil homomorphism
A dg-category of representations of Tg
A dg-category of modules over C(G)
Ingredients of the proof
Chen's iterated integrals
Bott-Shulman-Stasheff algebra
Ax de Rham theorem for classifying spaces
The non-commutative Weil algebra
The Moore Loop space
Example 1: The trivial representation
Example II: Free loop space of BG