2010•Forum MathematicumRequires access

Notes on 1-parameter formal deformations of Hom-associative and Hom-Lie algebras

Abdenacer Makhlouf, Sergei D. Silvestrov

Abstract

The aim of this paper is to extend to Hom-algebra structures the theory of 1-parameter formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras by Nijenhuis and Richardson. In this paper, formal deformations of Hom-associative and Hom-Lie algebras are studied. The first groups of a deformation cohomology are constructed and several examples of deformations are given. We also provide families of Hom-Lie algebras deforming Lie algebra 𝔰𝔩 2 (𝕂) and describe as formal deformations the q -deformed Witt algebra and Jackson 𝔰𝔩 2 (𝕂).

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Key concepts: Mathematics, Non-associative algebra, Associative property, Cohomology, Algebra over a field, Lie algebra, Pure mathematics, Universal enveloping algebra