Universal Coacting Hopf algebra of a Finite dimensional Lie-Yamaguti algebra

TL;DR

This paper constructs a universal coacting Hopf algebra for finite-dimensional Lie-Yamaguti algebras, extending Hopf algebra theory and providing tools for automorphism and grading classification.

Contribution

It introduces a universal algebra with a bialgebra structure for Lie-Yamaguti algebras, including a representation-theoretic perspective and applications to automorphisms and gradings.

Findings

Universal coacting Hopf algebra constructed for Lie-Yamaguti algebras

Characterization of automorphism groups of these algebras

Classification of all abelian group gradings

Abstract

M. E. Sweedler first constructed a universal Hopf algebra of an algebra. It is known that the dual notions to the existing ones play a dominant role in Hopf algebra theory. Yu. I. Manin and D. Tambara introduced the dual notion of Sweedler's construction in separate works. In this paper, we construct a universal algebra for a finite-dimensional Lie-Yamaguti algebra. We demonstrate that this universal algebra possesses a bialgebra structure, leading to a universal coacting Hopf algebra for a finite-dimensional Lie-Yamaguti algebra. Additionally, we develop a representation-theoretic version of our results. As an application, we characterize the automorphism group and classify all abelian group gradings of a finite-dimensional Lie-Yamaguti algebra.