Quasi-Lie algebras

Larsson, Daniel; Silvestrov, Sergei

doi:10.1090/conm/391/07333

Cited by 118 publications

(112 citation statements)

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“…Hom-Lie algebras were introduced in [3] as a tool in understanding the structure and constructions of q-deformations of the Witt and the Virasoro algebras within the general framework of quasi-Lie algebras and quasi-Hom-Lie algebras introduced in [6,7,13]. Hom-associative algebras, as an analogue and generalization of associative algebras for Hom-Lie algebras, have been introduced in [9].…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On Identities in Hom-Malcev Algebras

Issa¹

2015

International Electronic Journal of Algebra

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On Identities in Hom-Malcev Algebras

Issa¹

2015

International Electronic Journal of Algebra

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“…We summarize in the following the ungraded definitions of Hom-associative, Hom-Leibniz and Hom-Lie algebras (see [16]). Also, we recall the definition of quasi-Lie algebras (see [12]) which border Hom-Lie algebras, Hom-Lie superalgebras and color Lie algebras, as well as quasi-hom-Lie algebras appearing naturally in the study of σ -derivations. Definition 1.1.…”

Section: Hom-algebras and Graded Quasi-hom-lie Algebrasmentioning

confidence: 99%

“…A class of quasi-Leibniz algebras was introduced in [12] in connection to general quasi-Lie algebras following the standard Loday's conventions for Leibniz algebras (i.e. right Loday algebras).…”

Section: Hom-algebras and Graded Quasi-hom-lie Algebrasmentioning

confidence: 99%

“…The Hom-Lie algebras were discussed intensively in [9,[11][12][13] while the graded case was mentioned in [14]. Hom-associative algebras were introduced in [16], where it is shown that the commutator bracket of a Hom-associative algebra gives rise to a Hom-Lie algebra and where a classification of Hom-Lie admissible algebras is established.…”

Section: Introductionmentioning

confidence: 99%

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Hom-Lie superalgebras and Hom-Lie admissible superalgebras

2010

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MSC:17A70 17A30 16A03 17B68 Keywords:Hom-Lie superalgebra Hom-associative superalgebra Hom-Lie admissible superalgebra Lie admissible superalgebra q-Witt superalgebra The purpose of this paper is to study Hom-Lie superalgebras, that is a superspace with a bracket for which the superJacobi identity is twisted by a homomorphism. This class is a particular case of Γ -graded quasi-Lie algebras introduced by Larsson and Silvestrov. In this paper, we characterize Hom-Lie admissible superalgebras and provide a construction theorem from which we derive a one parameter family of Hom-Lie superalgebras deforming the orthosymplectic Lie superalgebra. Also, we prove a Z 2 -graded version of a Hartwig-Larsson-Silvestrov Theorem which leads us to a construction of a q-deformed Witt superalgebra.

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“…With this generalization of the Lie algebra, some q-deformations of the Witt and the Virasoro algebras have the structure of a Hom-Lie algebra [8]. Due to their close relationship with discrete and deformed vector fields and differential calculus [8,9,10], Hom-Lie algebras have been studied in broad areas [1,2,3,4,5,11,12,14,15,16,17,18,20].…”

Section: Introductionmentioning

confidence: 99%

Another approach to Hom-Lie bialgebras via Manin triples

Tao

Bai

Guo

2020

Communications in Algebra

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In this paper, we study Hom-Lie bialgebras by a new notion of the dual representation of a representation of a Hom-Lie algebra. Motivated by the essential connection between Lie bialgebras and Manin triples, we introduce the notion of a Hom-Lie bialgebra with emphasis on its compatibility with a Manin triple of Hom-Lie algebras associated to a nondegenerate symmetric bilinear form satisfying a new invariance condition. With this notion, coboundary Hom-Lie bialgebras can be studied without a skewsymmetric condition of r ∈ g ⊗ g, naturally leading to the classical Hom-Yang-Baxter equation whose solutions are used to construct coboundary Hom-Lie bialgebras. In particular, they are used to obtain a canonical Hom-Lie bialgebra structure on the double space of a Hom-Lie bialgebra. We also derive solutions of the classical Hom-Yang-Baxter equation from O-operators and Hom-left-symmetric algebras.

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Quasi-Lie algebras

Cited by 118 publications

References 0 publications

On Identities in Hom-Malcev Algebras

On Identities in Hom-Malcev Algebras

Hom-Lie superalgebras and Hom-Lie admissible superalgebras

Another approach to Hom-Lie bialgebras via Manin triples

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