Notes on Formal Deformations of Hom-associative and Hom-Lie Algebras

By Abdenacer Makhlouf, Sergei Silvestrov. 2007

132 citations recorded by the source corpus.

The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis-Richardson. We deal with Hom-associative and Hom-Lie algebras. We construct the first groups of a deformation cohomology and give several examples of deformations. We provide families of Hom-Lie algebras deforming Lie algebra $sl_2$ and describe as formal deformations the q-deformed Witt algebra and Jackson $sl_2$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/798ff08649f8ff34f0cab12aa5b9b973bd18d878

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Citations in the corpus

Hom-algebras and homology (2007, methodology): “ coeﬃcients. When applied to a Lie algebra L, our homology of L coincides with the usual Chevalley-Eilenberg homology of L [9]. The corresponding cohomology theory for Hom-Lie algebras was studied in [23]. 1.1. Organization. The rest of this paper is organized as follows. In the next section, basic ”
Hom-Lie Superalgebras and Hom-Lie admissible Superalgebras (2009, background): “See also [2, 3, 8, 19, 23, 25, 26] for other works on twisted algebraic structures.”
Hom-bialgebras and comodule Hom-algebras (2008, methodology): “ng the other way, from Hom-Lie algebras to Hom-associative algebras [7]. As in the classical case, the functor UHLie is the left adjoint of the functor HLie. Hom-algebras have been further studied in [5, 6, 8]. The purpose of this paper is to advance the theory of bialgebras in the Hom-algebra setti”
Yetter-Drinfeld modules for Hom-bialgebras (2013, methodology): “In what follows, we use terminology as in Yau’s paper [33], which is different from the original terminology in [27], [28] (where no extra assumption on the linear map α is made) and also different from Yau’s paper [34], where for instance the multiplicativity of the map α is emphasized by calling ””
On hom-algebras with surjective twisting (2009, background): “It was extended to hom-algebras in [4].”
Cohomology and Deformations of Hom-algebras (2010, methodology): “In the first Section we summarize the definitions of Hom-algebras of different type (see [1],[8],[14],[11]) and present some preliminary results on graded algebras (see [6], [10]).”
The Hom–Yang–Baxter equation, Hom–Lie algebras, and quasi-triangular bialgebras (2009, background): “Besides the sources cited above, other papers that discuss Hom–Lie algebras and related Hom-algebras include [8, 18, 19, 20, 27].”
On universal central extensions of Hom-Lie algebras (2012, background): “bracket operation [−,−] : L × L → L which satisfies the Hom-Leibniz identity [α(x), [y, z]] = [[x, y], α(z)] − [[x, z], α(y)], for all x, y, z ∈ L, and it was the subject of the recent papers [1, 10, 14, 15, 18].”
Generalization of n-ary Nambu algebras and beyond (2008, background): “The fundamentals of the formal deformation theory and associated cohomology structures for Hom-Lie algebras have been considered recently by the second and the third authors in [63].”
Representations of Hom-Lie Algebras (2010, background): “Makhlouf and Silvestrov studied the formal deformations of hom-Lie algebras in [9], where only 1-hom-cochains, 2-hom-cochains, 1-coboundary operator and 2-coboundary operator are defined without the help of any representation.”