The van Est homomorphism for strict Lie 2-groups

TL;DR

This paper develops a van Est map for strict Lie 2-groups, linking their cohomology to that of their Lie 2-algebras, and applies it to differentiate the Segal 2-form on loop groups.

Contribution

It introduces a van Est homomorphism for strict Lie 2-groups and demonstrates its cohomological isomorphism under certain conditions, advancing the understanding of higher Lie theory.

Findings

The van Est map induces cohomology isomorphisms under connectedness assumptions.

Application to differentiating the Segal 2-form on loop groups.

Establishment of a bridge between Lie 2-group cohomology and Lie 2-algebra cohomology.

Abstract

We construct a van Est map for strict Lie 2-groups from the Bott-Shulman-Stasheff double complex of the strict Lie 2-group to the Weil algebra of its associated strict Lie 2-algebra. We show that, under appropriate connectedness assumptions, this map induces isomorphisms in cohomology. As an application, we differentiate the Segal 2-form on the loop group.