Hom-bialgebras and comodule Hom-algebras

By Donald Yau. 2008

124 citations recorded by the source corpus.

We study Hom-bialgebras and objects admitting coactions by Hom-bialgebras. In particular, we construct a Hom-bialgebra M representing the functor of 2x2-matrices on Hom-associative algebras. Then we construct a Hom-algebra analogue of the affine plane and show that it is a comodule Hom-algebra over M in a suitable sense. It is also shown that the enveloping Hom-associative algebra of a Hom-Lie algebra is naturally a Hom-bialgebra.

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

View Paper: https://www.semanticscholar.org/paper/10ee9158eace36e203c85b029b8b9ab49a8deed7

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Citations in the corpus

The Hom–Yang–Baxter equation, Hom–Lie algebras, and quasi-triangular bialgebras (2009, methodology): “Dualizing and extending this procedure [19, 27 ], one can obtain Hom-coalgebras and Hom-bialgebras by deforming coalgebras and bialgebras, respectively, along endomorphisms.”
Rota–Baxter Hom-Lie–Admissible Algebras (2011, methodology): “The construction of the free Rota–Baxter G-Hom–associative algebra involves the combinatorial objects of decorated trees and is formally similar to the construction of the enveloping Homassociative algebras of Hom–Lie algebras [70, 72].”
Yetter-Drinfeld modules for Hom-bialgebras (2013, methodology): “In what follows, we use terminology as in Yau’s paper [33], which is different from the original terminology in [27], [28] (where no extra assumption on the linear map α is made) and also different from Yau’s paper [34], where for instance the multiplicativity of the map α is emphasized by calling ””
BiHom-associative algebras, BiHom-Lie algebras and BiHom-bialgebras (2015, methodology): “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].”
Monoidal Hom–Hopf Algebras (2009, background): “ of a Hom-bialgebra, it is not clear whether we should take diﬀerent endomorphisms describing the Hom-associativity and the Hom-coassociativity, see [8], where diﬀerent endomorphisms are allowed, and [12], where the two endomorphisms are the same. In [1], the authors compare all possible variations ”
Hom-algebras and homology (2007, background): “rs [26]. In fact, a unital version of U HLie(L) has the structure of a Hom-bialgebra [27, Theorem 3.12], generalizing the bialgebra structure on the usual enveloping algebra of a Lie algebra. Besides [21, 26, 27, 30, 31], Hom-algebras have been further studied in [2, 5, 8, 10, 11, 12, 17, 22, 23, 24”
Hom-algebras and Hom-coalgebras (2008, methodology): “In [ 47 ], it is shown that starting from an algebra and algebra endomorphism, one can construct a Homassociative algebra.”
Hom-Lie Superalgebras and Hom-Lie admissible Superalgebras (2009, background): “rsal enveloping Hom-associative algebra (see [22]). Dualizing Hom-associative algebras, one can deﬁne Hom-coassociative coalgebras, Hom-bialgebras and Hom-Hopf algebras (see [18, 20]). It is shown in [24] that the universal enveloping Hom-associative algebra carries a structure of Hom-bialgebra. See”