# Hom-bialgebras and comodule Hom-algebras

@article{Yau2008HombialgebrasAC, title={Hom-bialgebras and comodule Hom-algebras}, author={Donald Yau}, journal={arXiv: Rings and Algebras}, year={2008} }

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.

## 115 Citations

Infinitesimal Hom-bialgebras and Hom-Lie bialgebras

- Mathematics
- 2010

We study the Hom-type generalization of infinitesimal bialgebras, called infinitesimal Hom-bialgebras. In particular, we consider infinitesimal Hom-bialgebras arising from quivers, the sub-classes of…

Paradigm of Nonassociative Hom-algebras and Hom-superalgebras

- Mathematics
- 2010

The aim of this paper is to give a survey of nonassociative Hom-algebra and Hom-superalgebra structures. The main feature of these algebras is that the identities defining the structures are twisted…

2-Hom-Associative Bialgebras and Hom-Left Symmetric Dialgebras

- MathematicsTrends in Mathematics
- 2020

From the definition and properties of unital hom-associative algebras, and the use of the Kaplansky's constructions, we develop new algebraic structures called 2-hom-associative bialgebras,…

SOME RESULTS ABOUT HOM-COMODULE ALGEBRAS WITH HOM-HOPF MODULE STRUCTURE

- MathematicsInternational Electronic Journal of Algebra
- 2019

The main subject of this paper is Hom-comodule algebras with Hom-Hopf module structure. First, we give the factorization of a class of Hom- bialgebras, which is not only Hom-module coalgebras but…

Gerstenhaber–Schack cohomology for Hom-bialgebras and deformations

- Mathematics
- 2016

ABSTRACT Hom-bialgebras and Hom-Hopf algebras are generalizations of bialgebra and Hopf algebra structures, where associativity and coassociativity conditions are twisted by a homomorphism. The…

Hom-algebras and Hom-coalgebras

- Mathematics
- 2008

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…

On Twisted Smash Products of Monoidal Hom–Hopf Algebras

- Mathematics
- 2016

Let (H, α) be a monoidal Hom-Hopf algebra and (A, β) be an (H, α)-Hom-bimodule algebra. In this article, we first introduce the notion of a twisted Hom-smash product A☆H and then find some sufficient…

Crossed products of Hom-Hopf algebras

- Mathematics
- 2020

Let (H, α) be a Hom-Hopf algebra and (A, β) be a Hom-algebra. In this paper we will construct the Hom-crossed product (A#σH, β⊗α), and prove that the extension A ⊆ A#σH is actually a Hom-type cleft…

On Hom-type algebras

- Mathematics
- 2009

Hom-algebras are generalizations of algebras obtained using a twisting by a linear map. But there is a priori a freedom on where to twist.We enumerate here all the possible choices in the Lie and…

Variety of Hom-Sabinin Algebras and Related Algebra Subclasses

- MathematicsSpringer Proceedings in Mathematics & Statistics
- 2021

The purpose of this paper is to study Sabinin algebras of Hom-type. It is shown that Lie, Malcev, Bol and other algebras of Hom-type are naturally Sabinin algebras of Hom-type. To this end, we…

## References

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On Hom-algebra structures

- Mathematics
- 2006

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…

On Hom-type algebras

- Mathematics
- 2009

Hom-algebras are generalizations of algebras obtained using a twisting by a linear map. But there is a priori a freedom on where to twist.We enumerate here all the possible choices in the Lie and…

Hom-Hopf algebras

- Mathematics
- 2009

algebras, from of view of categories. This leads to the natural deﬁnition of monoidal Hom-algebras, Hom-coalgebras, etc. The authors construct a symmetric monoidal category, and then introduce…

HOM-ALTERNATIVE ALGEBRAS AND HOM-JORDAN ALGEBRAS

- Mathematics
- 2010

The main feature of Hom-algebras is that the identities deflning the structures are twisted by homomorphisms. The purpose of this paper is to introduce Hom-alternative algebras and Hom-Jordan…

Notes on Formal Deformations of Hom-associative and Hom-Lie Algebras

- Mathematics
- 2007

The aim of this paper is to extend to Hom-algebra structures the theory of formal deformations of algebras which was introduced by Gerstenhaber for associative algebras and extended to Lie algebras…

Hom-algebras and homology

- Mathematics
- 2007

Classes of $G$-Hom-associative algebras are constructed as deformations of $G$-associative algebras along algebra endomorphisms. As special cases, we obtain Hom-associative and Hom-Lie algebras as…

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

- Mathematics
- 2007

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…

The classical Hom-Yang-Baxter equation and Hom-Lie bialgebras

- Mathematics
- 2009

Motivated by recent work on Hom-Lie algebras and the Hom-Yang-Baxter equation, we introduce a twisted generalization of the classical Yang-Baxter equation (CYBE), called the classical Hom-Yang-Baxter…