Cohomology Characterizations of Diagonal Non-Abelian Extensions of Regular Hom-Lie Algebras

Rong Tang,Lina Song

doi:10.3390/sym9120297

Abstract

In this paper, first we show that under the assumption of the center of h being zero, diagonal non-abelian extensions of a regular Hom-Lie algebra g by a regular Hom-Lie algebra h are in one-to-one correspondence with Hom-Lie algebra morphisms from g to Out ( h ) . Then for a general Hom-Lie algebra morphism from g to Out ( h ) , we construct a cohomology class as the obstruction of existence of a non-abelian extension that induces the given Hom-Lie algebra morphism.

Highlights

The notion of a Hom-Lie algebra was introduced by Hartwig, Larsson, and Silvestrov in [1] as part of a study of deformations of the Witt and the Virasoro algebras
We show that if the center of h is zero and the short exact sequence related to derivations of Hom-Lie algebras is diagonal, diagonal non-abelian extensions of g by h correspond bijectively to Hom-Lie algebra morphisms from g to Out(h) (Theorem 2)
We show that the obstruction of the existence of a diagonal non-abelian extension of g by h that induces a given Hom-Lie algebra morphism from g to Out(h) is given by a cohomology class in H3 (g; Cen(h)) (Theorem 3)

Summary

Introduction

The notion of a Hom-Lie algebra was introduced by Hartwig, Larsson, and Silvestrov in [1] as part of a study of deformations of the Witt and the Virasoro algebras. We need to add some conditions on the short exact sequence related to derivations of Hom-Lie algebras Under these conditions, first we show that under the assumption of the center of h being zero, there is a one-to-one correspondence between diagonal non-abelian extensions of g by h and Hom-Lie algebra morphisms from g to Out(h). We show that if the center of h is zero and the short exact sequence related to derivations of Hom-Lie algebras is diagonal, diagonal non-abelian extensions of g by h correspond bijectively to Hom-Lie algebra morphisms from g to Out(h) (Theorem 2). We show that the obstruction of the existence of a diagonal non-abelian extension of g by h that induces a given Hom-Lie algebra morphism from g to Out(h) is given by a cohomology class in H3 (g; Cen(h)) (Theorem 3).

Preliminaries

Non-Abelian Extensions of Hom-Lie Algebras

Classification of Diagonal Non-Abelian Extensions of Hom-Lie Algebras

Conclusions