On the structure of split involutive Hom-Lie color algebras

Valiollah Khalili

doi:10.33044/revuma.v60n1a05

Abstract

In this paper we study the structure of arbitrary split involutive regular Hom-Lie color algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split involutive regular Hom-Lie color algebra L is of the form L=U⊕∑[α∈Π/∼I[α], with U a subspace of the involutive abelian subalgebra H and any I[α], a well-described involutive ideal of L, satisfying [I[α], I[β]]=0 if [α]≠[β]. Under certain conditions, in the case of L being of maximal length, the simplicity of the algebra is characterized and it is shown that L is the direct sum of the family of its minimal involutive ideals, each one being a simple split involutive regular Hom-Lie color algebra. Finally, an example will be provided to characterise the inner structure of split involutive Hom-Lie color algebras.

Highlights

The notion of Lie color algebras was introduced as generalized Lie algebras in 1960 by Ree in [11]
In 2012, Yuan [20] introduced the notion of a Hom-Lie color algebra which can be considered as an extension of Hom-Lie superalgebras to Λ-graded algebras, where Λ is any additive abelian group
For a physical system, which displays a symmetry of Lie algebra L, it is interesting to know in detail the structure of the split decomposition, because its roots can be seen as certain eigenvalues which are the additive quantum numbers characterizing the state of such a system

Summary

Introduction

The notion of Lie color algebras was introduced as generalized Lie algebras in 1960 by Ree in [11]. In 2012, Yuan [20] introduced the notion of a Hom-Lie color algebra which can be considered as an extension of Hom-Lie superalgebras to Λ-graded algebras, where Λ is any additive abelian group. Our goal in this work is to study the structure of arbitrary split involutive regular Hom-Lie color algebras by the techniques of connection of roots. Throughout this paper, split involutive regular Hom-Lie color algebras L are considered of arbitrary dimension and over an arbitrary base field F, with characteristic zero.

Preliminaries

Connections of roots and decompositions

The simple components

Example