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14 pages, 277 KiB

Open AccessArticle

The Ore Extension of Group-Cograded Hopf Coquasigroups

by Lingli Zhu, Bingbing Jin, Huili Liu and Tao Yang

Mathematics 2023, 11(17), 3703; https://doi.org/10.3390/math11173703 - 28 Aug 2023

Viewed by 518

The aim of this paper is the Ore extension of group-cograded Hopf coquasigroups. This paper first shows a categorical interpretation and some examples of group-cograded Hopf coquasigroups, and then provides a necessary and sufficient condition for the Ore extension of group-cograded Hopf coquasigroups [...] Read more.

The aim of this paper is the Ore extension of group-cograded Hopf coquasigroups. This paper first shows a categorical interpretation and some examples of group-cograded Hopf coquasigroups, and then provides a necessary and sufficient condition for the Ore extension of group-cograded Hopf coquasigroups to be group-cograded Hopf coquasigroups. Finally, a certain isomorphism between Ore extensions are considered. Full article

(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)

7 pages, 247 KiB

Open AccessArticle

Gorenstein Flat Modules of Hopf-Galois Extensions

by Qiaoling Guo, Tingting Shan, Bingliang Shen and Tao Yang

Mathematics 2023, 11(11), 2542; https://doi.org/10.3390/math11112542 - 1 Jun 2023

Viewed by 786

Abstract

Let

A / B

be a right H-Galois extension over a semisimple Hopf algebra H. The purpose of this paper is to give the relationship of Gorenstein flat dimensions between the algebra A and its subalgebra B, and obtain that [...] Read more.

Let

A / B

be a right H-Galois extension over a semisimple Hopf algebra H. The purpose of this paper is to give the relationship of Gorenstein flat dimensions between the algebra A and its subalgebra B, and obtain that the global Gorenstein flat dimension and the finitistic Gorenstein flat dimension of A is no more than that of B. Then the problem of preserving property of Gorenstein flat precovers for the Hopf-Galois extension will be studied. Finally, more relations for the crossed products and smash products will be obtained as applications. Full article

(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)

16 pages, 300 KiB

Open AccessArticle

A Duality Theorem for Hopf Quasimodule Algebras

by Huaiwen Guo and Shuanhong Wang

Mathematics 2023, 11(6), 1401; https://doi.org/10.3390/math11061401 - 14 Mar 2023

Viewed by 2532

Abstract

In this paper, we introduce and study two smash products

A ★ H

for a left H-quasimodule algebra A over a Hopf quasigroup H over a field

K

and

B # U

for a coquasi U-module algebra B over a Hopf [...] Read more.

In this paper, we introduce and study two smash products

A ★ H

for a left H-quasimodule algebra A over a Hopf quasigroup H over a field

K

and

B # U

for a coquasi U-module algebra B over a Hopf coquasigroup U, respectively. Then, we prove our duality theorem

(A ★ H) # H^{*} ≅ A \otimes (H # H^{*}) ≅ A \otimes M_{n} (K) ≅ M_{n} (A)

in the setting of a Hopf quasigroup H of dimension n. As an application of our result, we consider a special case of a finite quasigroup. Full article

(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)

15 pages, 316 KiB

Open AccessArticle

On the BiHom-Type Nonlinear Equations

by Hui Wu and Xiaohui Zhang

Mathematics 2022, 10(22), 4360; https://doi.org/10.3390/math10224360 - 19 Nov 2022

Viewed by 984

Abstract

In this paper, the Heisenberg doubles and Long dimodules of a BiHom-Hopf algebra are introduced. Then, we discussed the relation between BiHom-Hopf equation and BiHom-pentagon equation, and we obtain the solutions of BiHom-Hopf equation from Heisenberg doubles. We also showed that the parametric [...] Read more.

In this paper, the Heisenberg doubles and Long dimodules of a BiHom-Hopf algebra are introduced. Then, we discussed the relation between BiHom-Hopf equation and BiHom-pentagon equation, and we obtain the solutions of BiHom-Hopf equation from Heisenberg doubles. We also showed that the parametric generalized Long dimodules can provide the solutions of BiHom-Yang-Baxter equation and generalized

D

-equation. Full article

(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)

24 pages, 379 KiB

Open AccessArticle

Group-Graded By-Product Construction and Group Double Centralizer Properties

by Senlin Zhang and Shuanhong Wang

Mathematics 2022, 10(16), 2943; https://doi.org/10.3390/math10162943 - 15 Aug 2022

Viewed by 963

Abstract

For a group

π

with unit e, we introduce and study the notion of a

π

-graded Hopf algebra. Then we introduce and construct a new braided monoidal category

_{H}^{H_{e}} {YD}_{π}

over a

π

-graded Hopf algebra H. [...] Read more.

For a group

π

with unit e, we introduce and study the notion of a

π

-graded Hopf algebra. Then we introduce and construct a new braided monoidal category

_{H}^{H_{e}} {YD}_{π}

over a

π

-graded Hopf algebra H. We introduce the notion of a

π

-double centralizer property and investigate this property by studying a braided

π

-graded Hopf algebra

U (g l_{n} (V)) ⋉_{π} H

, where V is an n-dimensional vector space in

_{H}^{H_{e}} {YD}_{π}

and

U (g l_{n} (V))

is the braided universal enveloping algebra of

g l_{n} (V)

which is not the usual Hopf algebra. Finally, some examples and special cases are given. Full article

(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)

17 pages, 299 KiB

Open AccessArticle

3-Hom–Lie Yang–Baxter Equation and 3-Hom–Lie Bialgebras

by Shuangjian Guo, Shengxiang Wang and Xiaohui Zhang

Mathematics 2022, 10(14), 2485; https://doi.org/10.3390/math10142485 - 17 Jul 2022

Cited by 1 | Viewed by 1056

Abstract

In this paper, we first introduce the notion of a 3-Hom–Lie bialgebra and give an equivalent description of the 3-Hom–Lie bialgebras, the matched pairs and the Manin triples of 3-Hom–Lie algebras. In addition, we define

O

-operators of 3-Hom–Lie algebras and construct solutions [...] Read more.

In this paper, we first introduce the notion of a 3-Hom–Lie bialgebra and give an equivalent description of the 3-Hom–Lie bialgebras, the matched pairs and the Manin triples of 3-Hom–Lie algebras. In addition, we define

O

-operators of 3-Hom–Lie algebras and construct solutions of the 3-Hom–Lie Yang–Baxter equation in terms of

O

-operators and 3-Hom–pre-Lie algebras. Finally, we show that a 3-Hom–Lie algebra has a phase space if and only if it is sub-adjacent to a 3-Hom–pre-Lie algebra. Full article

(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)

15 pages, 263 KiB

Open AccessArticle

The Classical Hom–Leibniz Yang–Baxter Equation and Hom–Leibniz Bialgebras

by Shuangjian Guo, Shengxiang Wang and Xiaohui Zhang

Mathematics 2022, 10(11), 1920; https://doi.org/10.3390/math10111920 - 3 Jun 2022

Viewed by 1185

Abstract

In this paper, we first introduce the notion of Hom–Leibniz bialgebras, which is equivalent to matched pairs of Hom–Leibniz algebras and Manin triples of Hom–Leibniz algebras. Additionally, we extend the notion of relative Rota–Baxter operators to Hom–Leibniz algebras and prove that there is [...] Read more.

In this paper, we first introduce the notion of Hom–Leibniz bialgebras, which is equivalent to matched pairs of Hom–Leibniz algebras and Manin triples of Hom–Leibniz algebras. Additionally, we extend the notion of relative Rota–Baxter operators to Hom–Leibniz algebras and prove that there is a Hom–pre-Leibniz algebra structure on Hom–Leibniz algebras that have a relative Rota–Baxter operator. Finally, we study the classical Hom–Leibniz Yang–Baxter equation on Hom–Leibniz algebras and present its connection with the relative Rota–Baxter operator. Full article

(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)

17 pages, 288 KiB

Open AccessArticle

Yetter–Drinfeld Modules for Group-Cograded Hopf Quasigroups

by Huili Liu, Tao Yang and Lingli Zhu

Mathematics 2022, 10(9), 1388; https://doi.org/10.3390/math10091388 - 21 Apr 2022

Cited by 2 | Viewed by 1116

Abstract

Let H be a crossed group-cograded Hopf quasigroup. We first introduce the notion of p-Yetter–Drinfeld quasimodule over H. If the antipode of H is bijective, we show that the category

Y D Q (H)

of Yetter–Drinfeld quasimodules over H [...] Read more.

Let H be a crossed group-cograded Hopf quasigroup. We first introduce the notion of p-Yetter–Drinfeld quasimodule over H. If the antipode of H is bijective, we show that the category

Y D Q (H)

of Yetter–Drinfeld quasimodules over H is a crossed category, and the subcategory

Y D (H)

of Yetter–Drinfeld modules is a braided crossed category. Full article

(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)

23 pages, 388 KiB

Open AccessArticle

Generalized Hopf–Ore Extensions of Hopf Group-Coalgebras

by Ding-Guo Wang and Xing Wang

Mathematics 2022, 10(7), 1167; https://doi.org/10.3390/math10071167 - 3 Apr 2022

Viewed by 1035

Abstract

In this paper, we introduce the concept of a generalized Hopf–Ore extension of a Hopf group-coalgebra and give the necessary and sufficient conditions for the Ore extension of a Hopf group-coalgebra to be a Hopf group-coalgebra. Moreover, an isomorphism theorem on generalized Hopf [...] Read more.

In this paper, we introduce the concept of a generalized Hopf–Ore extension of a Hopf group-coalgebra and give the necessary and sufficient conditions for the Ore extension of a Hopf group-coalgebra to be a Hopf group-coalgebra. Moreover, an isomorphism theorem on generalized Hopf group-coalgebra Ore extensions is given and specific cases in a simple type called special Hopf group-coalgebra Ore extensions are also considered. Our results are generalizations of Hopf–Ore extensions on Hopf algebras and also are very useful to construct new Hopf group-coalgebras. Full article

(This article belongs to the Special Issue Hopf-Type Algebras, Lie Algebras, Quantum Groups and Related Topics)

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