# Spatial realization of a Lie algebra and Bar construction of a group

## TL;DR

This paper establishes an isomorphism between the spatial realization of a rational Lie algebra and the simplicial bar construction of a group, linking algebraic and topological structures through the Baker-Campbell-Hausdorff product.

## Contribution

It demonstrates a novel isomorphism connecting the spatial realization of a Lie algebra with the bar construction of a group, advancing the understanding of their algebraic-topological relationship.

## Key findings

- Spatial realization of a Lie algebra is isomorphic to the bar construction of a group.
- The isomorphism is established via the Baker-Campbell-Hausdorff product.
- The work bridges Lie algebra structures with simplicial group constructions.

## Abstract

We prove that the spatial realization of a rational complete Lie algebra $L$, concentrated in degree 0, is isomorphic to the simplicial bar construction on the group, obtained from the Baker-Campbell-Hausdorff product on $L$.