On Hom-algebra structures

By Abdenacer Makhlouf, Sergei Silvestrov. 2006

506 citations recorded by the source corpus.

A Hom-algebra structure is a multiplication on a vector space where the structure is twisted by a homomorphism. The structure of Hom-Lie algebra was introduced by Hartwig, Larsson and Silvestrov and extended by Larsson and Silvestrov to quasi-hom Lie and quasi-Lie algebras. In this paper we introduce and study Hom-associative, Hom-Leibniz, and Hom-Lie admissible algebraic structures which generalize the well known associative, Leibniz and Lie admissible algebras. Also, we characterize the flexible Hom-algebras in this case. We also explain some connections between Hom-Lie algebras and Santilli's isotopies of associative and Lie algebras.

Subfields: Algebra and Number Theory

View Paper: https://www.semanticscholar.org/paper/57aff195ab49dff3e7e50b39835665df4b76ac2d

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In Algebra and Number Theory

1. Abdenacer Makhlouf
2. Donald Yau
3. Florin Panaite
4. Y. Sheng
5. F. Ammar
6. Aron Gohr
7. S. Benayadi
8. Sergei Silvestrov

Citations in the corpus

• Unital algebras of Hom-associative type and surjective or injective twistings (2009, methodology): “Also, in [15], a generalization of Hom-associative algebras, G-Hom-associative algebras, and their subclasses left symmetric Hom-algebras, Hom-pre-Lie algebras, or right symmetric Hom-algebras, and two other new classes of Hom-algebras have been introduced and G-Hom-associative algebras”
• On universal central extensions of Hom-Lie algebras (2012, unknown): “In this way, the notion of Hom-Leibniz algebra was first introduced in [15] as Kvector spaces L together with a linear map α :L → L, endowed with a bilinear 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 ∈”
• Hom-algebras and Hom-coalgebras (2008, methodology): “acx2. We call a triple (V,µ,α)Hom-Lie admissible algebraif the commutator defined for x,y ∈V by [x,y]=µ(x,y)−µ(y,x)defines a Hom-Lie algebra (V,[·,·],α). The Hom-Lie admissible algebras were studied in [39], where we have shown among other things, that the Hom-associative algebras are Hom-Lie admissib”
• Rota–Baxter Hom-Lie–Admissible Algebras (2011, methodology): “fRota-BaxterHom-associativealgebrasand ofHom-(tri)dendriform algebras. 2. Hom-Lie admissible and G-Hom-associative algebras In this Section we recall the main result of Hom-Lie-admissible algebras in [58] and summarize the definitions and some properties of G-Hom-associativealgebras. The latter are t”
• Hom-algebras and homology (2007, background): “Note that in [21], α was not required to be multiplicative.”
• Representations of Hom-Lie Algebras (2010, methodology): “Because of close relation to discrete and deformed vector fields and differential calculus [5, 7, 8], hom-Lie algebras are widely studied recently [1, 2, 9, 10, 12, 13, 14].”
• HOM-ALTERNATIVE ALGEBRAS AND HOM-JORDAN ALGEBRAS (2009, methodology): “This process was introduced in [30] for Lie and associative algebras and more generally to G-associative algebras (see [18] for this class of algebras) and generalized to coalgebras in [21], [19] and to n-ary algebras of Lie and associative types in [3].”
• Hom-Lie Algebras with Symmetric Invariant NonDegenerate Bilinear Forms (2010, methodology): “e a give a structure theorem of centerless involutive quadratic Hom-Lie algebra. 1. Preliminaries In the following we summarize the definitions of Hom-Lie and Hom-associative algebraic structures (see [30]) generalizing the well known Lie and associative algebras. Also we define the notion of modules ”
• Yetter-Drinfeld modules for Hom-bialgebras (2013, methodology): “It was shown in [27] that a commutator of a Homassociative algebra gives rise to a Hom-Lie algebra; the construction of the free Hom-associative”
• BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras (2015, unknown): “Key words: BiHom-associative algebra; BiHom-Lie algebra; BiHom-bialgebra; representation; twisting; smash product 2010 Mathematics Subject Classification: 17A99; 18D10; 16T99”
• Hom-bialgebras and comodule Hom-algebras (2008, methodology): “It is proved in [18] that a Hom-associative algebra A gives rise to a Hom-Lie algebra HLie(A) via the commutator bracket construction, i.”
• Enveloping algebras of Hom-Lie algebras (2007, methodology): “In [8], Makhlouf and Silvestrov introduced the notion of a Hom-associative algebra (A,µ, α), in which the binary operation µ satisfies an α-twisted version of associativity.”