Hom-algebras and Hom-coalgebras

By Abdenacer Makhlouf, Sergei Silvestrov. 2008

285 citations recorded by the source corpus.

The aim of this paper is to develop the theory of Hom-coalgebras and related structures. After reviewing some key constructions and examples of quasi-deformations of Lie algebras involving twisted derivations and giving rise to the class of quasi-Lie algebras incorporating Hom–Lie algebras, we describe the notion and some properties of Hom-algebras and provide examples. We introduce Hom-coalgebra structures, leading to the notions of Hom-bialgebra and Hom–Hopf algebras, and prove some fundamental properties and give examples. Finally, we define the concept of Hom–Lie admissible Hom-coalgebra and provide their classification based on subgroups of the symmetric group.

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

View Paper: https://www.semanticscholar.org/paper/270a697f3b71f18f47863fcf1cb6c15dcd8f9759

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Citations in the corpus

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].”
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].”
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].”
Rota–Baxter Hom-Lie–Admissible Algebras (2011, background): “In [61] and [63], the theory of Hom-coalgebras and related structures are developed.”
The Hom–Yang–Baxter equation, Hom–Lie algebras, and quasi-triangular bialgebras (2009, methodology): “Besides the sources cited above, other papers that discuss Hom-Lie algebras and related Hom-algebras include [7, 17, 18, 19, 26].”
Monoidal Hom–Hopf Algebras (2009, methodology): “In the definition of a Hom-bialgebra, it is not clear whether we should take different endomorphisms describing the Hom-associativity and the Hom-coassociativity, see [10], where different endomorphisms are allowed, and [14], where the two endomorphisms are”
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 [18, 20]).”
Yetter-Drinfeld modules for Hom-bialgebras (2013, background): “6 ([29], [30]) A Hom-bialgebra is a quadruple (H,μ,∆, α), in which (H,μ, α) is a Hom-associative algebra, (H,∆, α) is a Hom-coassociative coalgebra and moreover ∆ is a morphism of Hom-associative algebras.”
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].”