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.

In this paper the universal enveloping algebra of color hom-Lie algebras is studied. A construction of the free involutive hom-associative color algebra on a hom-module is described and applied to obtain the universal enveloping algebra of an involutive hom-Lie color algebra. Finally, the construction is applied to obtain the well-known Poincaré-Birkhoff-Witt theorem for Lie algebras to the enveloping algebra of an involutive color hom-Lie algebra. of abstract quasi-Lie algebras and subclasses of quasi-Hom-Lie algebras and Hom-Lie algebras as well as their general colored (graded) counterparts in [25,36,39,58,37]. These generalized Lie algebra structures with (graded) twisted skew-symmetry and twisted Jacobi conditions by linear maps are tailored to encompass within the same algebraic framework such quasi-deformations and discretizations of Lie algebras of vector fields using σ-derivations, describing general descritizations and deformations of derivations with twisted Leibniz rule, and the well-known generalizations of Lie algebras such as color Lie algebras which are the natural generalizations of Lie algebras and Lie superalgebras.Quasi-Lie algebras are non-associative algebras for which the skew-symmetry and the Jacobi identity are twisted by several deforming twisting maps and also the Jacobi identity in quasi-Lie and quasi-Hom-Lie algebras in general contains six twisted triple bracket terms. Hom-Lie algebras is a special class of quasi-Lie algebras with the bilinear product satisfying the non-twisted skew-symmetry property as in Lie algebras, whereas the Jacobi identity contains three terms twisted by a single linear map, reducing to the Jacobi identity for ordinary Lie algebras when the linear twisting map is the identity map. Subsequently, hom-Lie admissible algebras have been considered in [45] where also the hom-associative algebras have been introduced and shown to be hom-Lie admissible natural generalizations of associative algebras corresponding to hom-Lie algebras. In [45], moreover several other interesting classes of hom-Lie admissible algebras generalising some non-associative algebras, as well as examples of finite-dimentional hom-Lie algebras have been described. Since these pioneering works [25,36,39,37,40,45], hom-algebra structures have become a popular area with increasing number of publications in various directions.Hom-Lie algebras, hom-Lie superalgebras and hom-Lie color algebras are important special classes of color (Γ-graded) quasi-Lie algebras introduced first by Larsson and Silvestrov in [39,37]. Hom-Lie algebras and hom-Lie superalgebras have been studied further in different aspects by Makhlouf, Silvestrov, Sheng, Ammar, Yau and other authors [62,61,60,45,48,46,47,68,12,44,69,64,65,66,38,51,52,57,59], and hom-Lie color algebras have been considered for example in [68,14,13,1]. In [4], the constructions of Hom-Lie and quasi-hom Lie algebras based on twisted discretizations of vector fields [25] and Hom-Lie admissible algebras have been extended to Hom-Lie superalgebras, a subclass...

In this paper, we deal with (non-abelian) extensions of a given hom-Lie superalgebra and find a cohomological obstacle to the existence of extensions of hom-Lie superalgebras. Moreover, the setting of covariant exterior derivatives, super connection, curvature and the Bianchi identity in differential geometry has been studied.

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