From simplicial Lie algebras and hypercrossed complexes to differential graded Lie algebras via 1-jets

TL;DR

This paper constructs a differential graded Lie algebra from a simplicial Lie algebra using 1-jet techniques, providing a geometric perspective on Quillen's algebraic adjunction between these structures.

Contribution

It introduces a geometric method to derive dg-Lie algebras from simplicial Lie algebras via 1-jet of classifying spaces, linking hypercrossed complexes to differential graded structures.

Findings

Explicit description of the differential and brackets in the dg-Lie algebra

Geometric interpretation of Quillen's algebraic construction

Construction applicable to simplicial Lie algebras with Moore complex of finite length

Abstract

Let g be a simplicial Lie algebra with Moore complex Ng of length k. Let G be the simplicial Lie group integrating g, which is simply connected in each simplicial level. We use the 1-jet of the classifying space of G to construct, starting from g, a Lie k-algebra L. The so constructed Lie k-algebra L is actually a differential graded Lie algebra. The differential and the brackets are explicitly described in terms (of a part) of the corresponding k-hypercrossed complex structure of Ng. The result can be seen as a geometric interpretation of Quillen's (purely algebraic) construction of the adjunction between simplicial Lie algebras and dg-Lie algebras.