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2007•Unpublished venueRequires access

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

Abdenacer Makhlouf, Sergei D. Silvestrov

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Abstract

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 sl2 and describe as formal deformations the q-deformed Witt algebra and Jackson sl2.

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What this paper is about

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 sl2 and describe as formal deformations the q-deformed Witt algebra and Jackson sl2.

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Available abstract

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 sl2 and describe as formal deformations the q-deformed Witt algebra and Jackson sl2.

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

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