Hom-Lie groups of a class of Hom-Lie algebra

TL;DR

This paper introduces the concept of Hom-Lie groups, establishes their relationship with Hom-Lie algebras, and demonstrates their application to the Toda lattice equation, expanding the understanding of algebraic structures in differential geometry.

Contribution

It defines Hom-Lie groups, shows their connection to Hom-Lie algebras, and applies these concepts to the Toda lattice, providing new insights into algebraic structures and their applications.

Findings

A connected component of $GL(V)$ is a Hom-Lie group.

There is a one-to-one correspondence between Hom-Lie groups and Hom-Lie algebras.

Hom-Lie group homomorphisms induce morphisms between Hom-Lie algebras.

Abstract

In this paper, the definition of Hom-Lie groups is given and one conntected component of Lie group $GL(V)$, which is not a subgroup of $GL(V)$, is a Hom-Lie group. More, we proved that there is a one-to-one relationship between Hom-Lie groups and Hom-Lie algebras $(\gl(V),[\cdot,\cdot]_\beta,\rm{Ad}_\beta)$. Next, we also proved that if there is a Hom-Lie group homomorphism, then, there is a morphism between their Hom-Lie algebras. Last, as an application, we use these results on Toda lattice equation.