Integration of the Baker-Campbell-Hausdorff product

TL;DR

This paper introduces a new group operation in complete differential graded Lie algebras that aligns with the Baker-Campbell-Hausdorff product, enabling applications in homotopy theory and solving a specific problem in dimension 4.

Contribution

It constructs a novel group operation compatible with the differential in differential graded Lie algebras and applies it to model the 4-simplex in homotopy theory.

Findings

Defines a group operation on $L_1$ compatible with the differential

Provides a Lie model for the 4-simplex in homotopy theory

Solves a problem in dimension 4 posed by Lawrence and Sullivan

Abstract

In an arbitrary complete differential graded Lie algebra, we construct a group operation $\bullet$ on $L_1$ such that the differential of the product of two elements is the Baker-Campbell-Hausdorff product of their differentials, i.e., $d(x\bullet y)=dx\ast dy$. We study some properties of this new structure and some applications, especially in homotopy theory, where this operation can be used to construct a Lie model for the 4-simplex. In particular, this solves, in dimension 4, a problem proposed by Lawrence and Sullivan.