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23 pages, 319 KiB

Open AccessArticle

Quasigroups, Braided Hopf (Co)quasigroups and Radford’s Biproducts of Quasi-Diagonal Type

by Yue Gu and Shuanhong Wang

Mathematics 2024, 12(21), 3384; https://doi.org/10.3390/math12213384 - 29 Oct 2024

Cited by 1 | Viewed by 934

Given the Yetter–Drinfeld category over any quasigroup and a braided Hopf coquasigroup in this category, we first mainly study the Radford’s biproduct corresponding to this braided Hopf coquasigroup. Then, we investigate Sweedler’s duality of this braided Hopf coquasigroup and show that this duality is also a braided Hopf quasigroup in the Yetter–Drinfeld category, generalizing the main result in a Hopf algebra case of Ng and Taft’s paper. Finally, as an application of our results, we show that the space of binary linearly recursive sequences is closed under the quantum convolution product of binary linearly recursive sequences. Full article

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 1265

Abstract

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.

(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

Cited by 1 | Viewed by 3096

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, 306 KiB

Open AccessArticle

A Morita-Takeuchi Context and Hopf Coquasigroup Galois Coextensions

by Huaiwen Guo and Shuanhong Wang

Symmetry 2023, 15(2), 551; https://doi.org/10.3390/sym15020551 - 18 Feb 2023

Viewed by 1557

Abstract

For H a Hopf quasigroup and C, a left quasi H-comodule coalgebra, we show that the smash coproduct

C ⋊ H

(as a symmetry of smash product) is linked to some quotient coalgebra

Q = C / C H^{* +}

[...] Read more.

For H a Hopf quasigroup and C, a left quasi H-comodule coalgebra, we show that the smash coproduct

C ⋊ H

(as a symmetry of smash product) is linked to some quotient coalgebra

Q = C / C H^{* +}

by a Morita-Takeuchi context (as a symmetry of Morita context). We use the Morita-Takeuchi setting to prove that for finite dimensional H, equivalent conditions for

C / Q

to be a Hopf quasigroup Galois coextension (as a symmetry of Galois extension). In particular, we consider a special case of quasigroup graded coalgebras as an application of our theory. Full article

17 pages, 328 KiB

Open AccessArticle

Hopf Quasigroup Galois Extensions and a Morita Equivalence

by Huaiwen Guo and Shuanhong Wang

Mathematics 2023, 11(2), 273; https://doi.org/10.3390/math11020273 - 5 Jan 2023

Cited by 2 | Viewed by 1413

Abstract

For H, a Hopf coquasigroup, and A, a left quasi-H-module algebra, we show that the smash product

A # H

is linked to the algebra of H invariants

A^{H}

by a Morita context. We use the Morita setting [...] Read more.

For H, a Hopf coquasigroup, and A, a left quasi-H-module algebra, we show that the smash product

A # H

is linked to the algebra of H invariants

A^{H}

by a Morita context. We use the Morita setting to prove that for finite dimensional H, there are equivalent conditions for

A / A^{H}

to be Galois parallel in the case of H finite dimensional Hopf algebra. Full article

(This article belongs to the Section A: Algebra and Logic)

23 pages, 317 KiB

Open AccessArticle

Multiplier Hopf Coquasigroup: Motivation and Biduality

by Tao Yang

Mathematics 2022, 10(21), 4006; https://doi.org/10.3390/math10214006 - 28 Oct 2022

Cited by 1 | Viewed by 1372

Abstract

Inspired by the multiplier Hopf algebra theory introduced by A. Van Daele, this paper introduces a new algebraic structure, a multiplier Hopf coquasigroup, by constructing the integral dual of an infinite-dimensional Hopf quasigroup with faithful integrals. Then, it shows that the biduality theorem [...] Read more.

(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra)

40 pages, 446 KiB

Open AccessArticle

A New Approach to Braided T-Categories and Generalized Quantum Yang–Baxter Equations

by Senlin Zhang and Shuanhong Wang

Mathematics 2022, 10(6), 968; https://doi.org/10.3390/math10060968 - 17 Mar 2022

Cited by 1 | Viewed by 2000

Abstract

We introduce and study a large class of coalgebras (possibly (non)coassociative) with group-algebraic structures Hopf (non)coassociative group-algebras. Hopf (non)coassociative group-algebras provide a unifying framework for classical Hopf algebras and Hopf group-algebras and Hopf coquasigroups. We introduce and discuss the notion of a quasitriangular Hopf (non)coassociative π-algebra and show some of its prominent properties, e.g., antipode S is bijective. As an application of our theory, we construct a new braided T-category and give a new solution to the generalized quantum Yang–Baxter equation. Full article

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