# Differential graded Lie groups and their differential graded Lie algebras

## TL;DR

This paper explores the integration process between differential graded Lie algebras and differential graded Lie groups, establishing a correspondence using graded Hopf algebras and Harish-Chandra pairs.

## Contribution

It introduces the category of differential graded Lie groups, constructs the association with differential graded Lie algebras, and develops the theoretical framework for their integration.

## Key findings

- Established a correspondence between DGLAs and DGLGs.
- Defined the category of differential graded Lie groups.
- Used graded Hopf algebras and Harish-Chandra pairs in the construction.

## Abstract

In this paper we discuss the question of integrating differential graded Lie algebras (DGLA) to differential graded Lie groups (DGLG). We first recall the classical problem of integration in the context, and present the construction for (non-graded) differential Lie algebras. Then, we define the category of differential graded Lie groups and study its properties. We show how to associate a differential graded Lie algebra to every differential graded Lie group and vice-versa. For the DGLA $\to$ DGLG direction, the main ``tools'' are graded Hopf algebras and Harish-Chandra pairs (HCP) -- we define the category of graded and differential graded HCPs and explain how those are related to the desired construction. We describe some near at hand examples and mention possible generalizations.