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SIGMA 16 (2020), 137, 22 pages arXiv:1904.06515
https://doi.org/10.3842/SIGMA.2020.137
Hom-Lie Algebras and Hom-Lie Groups, Integration and Differentiation
Jun Jiang a, Satyendra Kumar Mishra b and Yunhe Sheng a
a) Department of Mathematics, Jilin University, Changchun, Jilin Province, 130012, China
b) Statistics and Mathematics Unit, Indian Statistical Institute Bangalore, India
Received June 01, 2020, in final form December 10, 2020; Published online December 17, 2020
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 ($\mathsf{Hexp}$) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this $\mathsf{Hexp}$ 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.
Key words: Hom-Lie algebra; Hom-Lie group; derivation; automorphism; integration.
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