Deformations of Lie brackets and representations up to homotopy

TL;DR

This paper develops a functorial approach to differentiating representations up to homotopy, proves a van Est isomorphism theorem, and confirms a conjecture on Lie bracket deformations.

Contribution

It introduces a functorial differentiation method for representations up to homotopy and proves a van Est isomorphism theorem related to Lie bracket deformations.

Findings

Established a van Est type isomorphism theorem.

Proved a conjecture of Crainic and Moerdijk on Lie bracket deformations.

Demonstrated functorial differentiation of representations up to homotopy.

Abstract

We show that representations up to homotopy can be differentiated in a functorial way. A van Est type isomorphism theorem is established and used to prove a conjecture of Crainic and Moerdijk on deformations of Lie brackets.