Hom-Lie Admissible Hom-coalgebras and Hom-Hopf Algebras

By Abdenacer Makhlouf, Sergei Silvestrov. 2007

189 citations recorded by the source corpus.

The aim of this paper is to generalize the concept of Lie-admissible coalgebra introduced by Goze and Remm to Hom-coalgebras and to introduce Hom-Hopf algebras with some properties. These structures are based on the Hom-algebra structures introduced by the authors.

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

View Paper: https://www.semanticscholar.org/paper/7ddb6a27f66815d965ce1f463f9973891bda9d26

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Citations in the corpus

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 Superalgebras and Hom-Lie admissible Superalgebras (2009, background): “Dualizing Hom-associative algebras, one can define Hom-coassociative coalgebras, Hom-bialgebras and Hom-Hopf algebras (see [16, 18]).”
Unital algebras of Hom-associative type and surjective or injective twistings (2009, background): “On the dual side, Hom-coalgebras, Hom-bialgabras, and Hom-Hopf algebras have been considered for the first time by Makhlouf and Silvestrov in [16] in 2007 and further studied by the same authors in [17] and also by Yau in [21].”
Hom-bialgebras and comodule Hom-algebras (2008, background): “Hom-type analogues of Hopf algebras have been studied in [19].”
Paradigm of Nonassociative Hom-algebras and Hom-superalgebras (2010, methodology): “This process was introduced in [45] for Lie and associative algebras and more generallyG-associative algebras (see [32] for this class of algebras) and generalized to coalgebras in [35], [33] and to n-ary algebras of Lie and associative types in [5].”
The Hom–Yang–Baxter equation, Hom–Lie algebras, and quasi-triangular bialgebras (2009, background): “Besides the sources cited above, other papers that discuss Hom-Lie algebras and related Hom-algebras include [7, 17, 18, 19, 26].”
Generalization of n-ary Nambu algebras and beyond (2008, methodology): “In [62] and [64], we developed the theory of Hom-coalgebras and related structures.”
Hom-algebras and homology (2007, background): “Besides [21, 26, 27, 30, 31], Hom-algebras have been further studied in [2, 5, 8, 10, 11, 12, 17, 22, 23, 24, 29, 32, 33, 34].”
BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras (2015, background): “In the last years, many concepts and properties from classical algebraic theories have been extended to the framework of Hom-structures, see for instance [2, 3, 7, 8, 11, 16, 20, 22, 25, 26, 27, 31, 30].”
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 ∈”