Cohomology and Deformations of Hom-algebras

By F. Ammar, Zeyneb Ejbehi, Abdenacer Makhlouf. Journal of Lie theory, 2010

208 citations recorded by the source corpus.

The purpose of this paper is to define cohomology structures on Hom-associative algebras and Hom-Lie algebras. The first and second coboundary maps were introduced by Makhlouf and Silvestrov in the study of one-parameter formal deformations theory. Among the relevant formulas for a generalization of Hochschild cohomology for Hom-associative algebras and a Chevalley-Eilenberg cohomology for Hom-Lie algebras, we define Gerstenhaber bracket on the space of multilinear mappings of Hom-associative algebras and Nijenhuis-Richardson bracket on the space of multilinear mappings of Hom-Lie algebras. Also we enhance the deformations theory of this Hom-algebras by studying the obstructions.

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

View Paper: https://www.semanticscholar.org/paper/55438130fe1255fafe1d1c404467b2e0ea7ffa7a

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Citations in the corpus

Yetter-Drinfeld modules for Hom-bialgebras (2013, background): “For instance, representation theory, cohomology and deformation theory for Hom-associative algebras and Hom-Lie algebras have been developed in [2, 28, 31].”
Rota–Baxter Hom-Lie–Admissible Algebras (2011, background): “The fundamentals of the formal deformation theory and associated cohomology structures for Hom–Lie algebras have been considered initially in [62] and completed in [6].”
Representations of Hom-Lie Algebras (2010, methodology): “Remark 6.3 In [ 1 ], the authors obtain a cochain complex for multiplicative hom-Lie algebra independently.”
On universal central extensions of Hom-Lie algebras (2012, background): “In particular, a (co)homology theory for Hom-Lie algebras, which generalizes the ChevalleyEilenberg (co)homology for a Lie algebra, was the subject of [1, 2, 3, 10, 12].”