Geometric aspects of extensions of hom-Lie superalgebras

Abstract

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.

Extensions of hom-Lie color algebras

“…Then it can be considered as a connection in the principal bundle π, and the h-valued 2-form ρ as its curvature. This geometric concept example is a very good guideline for our approach which is dealt with in [8,9]. It works also for color hom-Lie algebras.…”

Z-graded Hom-Lie Superalgebras

“…In [18,20,22,27], S. Silvestrov et al introduced the general quasi-Lie algebras and including as special cases the color hom-Lie algebras [5,7,8] and in particular hom-Lie superalgebras. Recently, different features of hom-Lie superalgebras has been studied by authors in [3,4,6,25,29]. On the other hand, we have the graded structure on Lie algebras which were discussed by authors.…”

Simply Complete Hom-Lie Superalgebras and Decomposition of Complete Hom-Lie Superalgebras

“…The Hom-Lie superalgebras and the more general color quasi-Lie algebras provide new general parametric families of non-associative structures, extending and interpolating on the fundamental level of defining identities between the Lie algebras, Lie superalgebras, color Lie algebras and some other important related non-associative structures, their deformations and discretizations, in the special interesting ways which may be useful for unification of models of classical and quantum physics, geometry and symmetry analysis, and also in algebraic analysis of computational methods and algorithms involving linear and non-linear discretizations of differential and integral calculi. Investigation of color hom-Lie algebras and hom-Lie superalgebras and n-ary generalizations have been further expanded recently in [1,2,7,8,[11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29]33,42,43,[48][49][50]60,61,64,65,[69][70][71][72][73]75].…”