# Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation

## TL;DR

This paper introduces the concept of Hom-Lie groups and explores their relationship with Hom-Lie algebras, including integration, the Hom-exponential map, and representations, extending classical Lie theory to the Hom-Lie setting.

## Contribution

It defines regular Hom-Lie groups, establishes their integrability, and develops the Hom-exponential map and representation theory, advancing the understanding of Hom-Lie structures.

## Key findings

- Every regular Hom-Lie algebra is integrable.
- Defined a Hom-exponential map with universality properties.
- Constructed examples including the Hom-Lie algebra of endomorphisms.

## Abstract

In this paper, we introduce the notion of a (regular) Hom-Lie group. We associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential (Hexp) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this Hexp map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra $(\mathfrak{gl}(V),[\cdot,\cdot],\mathsf{Ad})$, and the derivation Hom-Lie algebra of a Hom-Lie algebra.