Extensions of hom-Lie color algebras

Abstract

In this paper we study (non-Abelian) extensions of a given hom-Lie color algebra and provide a geometrical interpretation of extensions. In particular, we characterize an extension of a hom-Lie algebra g by another hom-Lie algebra h and we discuss the case where h has no center. We also deal with the setting of covariant exterior derivatives, Chevalley derivative, curvature and the Bianchi identity for the possible extensions in differential geometry. Moreover, we find a cohomological obstruction to the existence of extensions of hom-Lie color algebras, i. e. we show that in order to have an extendible hom-Lie color algebra, there should exist a trivial member of the third cohomology.M.S.C. 2010: 17B56, 17B75, 17B40.

Generalized Derivations and Rota-Baxter Operators of $$\varvec{n}$$-ary Hom-Nambu Superalgebras

“…Generalizations of derivations in connection with extensions and enveloping algebras of Hom-Lie color algebras and Hom-Lie superalgebras have been considered in [12,13,31,48]. Generalized derivations of multiplicative nary Hom-Ω color algebras have been studied in [36].…”

HNN-extension of involutive multiplicative Hom-Lie algebras

“…This approach has been extended to the enveloping algebras for color Hom-Lie algebras in [11,12]. Extensions of Hom-Lie superalgebras and Hom-Lie color algebras have been considered in [9,13]. Hom-associative Ore extensions have been considered in [29][30][31][32][33][34] The paper is organized as follows.…”

Simply complete hom-Lie superalgebras and decomposition of complete hom-Lie superalgebras

“…The authors studied quadratic hom-Lie algebras in [11]; representation theory, cohomology and homology theory in [2,30,32]. In [22,24,26,31], S. Silvestrov et al introduced the general quasi-Lie algebras and including as special cases the color hom-Lie algebras [6,8,9] and in particular hom-Lie superalgebras. Recently, different features of hom-Lie superalgebras has been studied by authors in [3,4,7,29,33].…”