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Symmetry
Special Issues
Hopf Algebras, Quantum Groups and Monoidal Categories
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Hopf Algebras, Quantum Groups and Monoidal Categories

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (25 November 2021) | Viewed by 2621

Special Issue Editors

Prof. Dr. Blas Torrecillas

E-Mail Website
Guest Editor
Campus Universitario, Universidad de Almería, Almería, Spain
Interests: noncommutative algebra; homological algebra; quantum groups; category theory
Prof. Dr. Claudia Menini

E-Mail Website
Guest Editor
University of Ferrara
Interests: abstract algebra; algebra; category theory; representation theory; topology

Special Issue Information

Dear Colleagues,

It is very well-known that group theory is the algebraic structure associated to symmetries. Hopf algebras, that generalized groups, models symmetries in a more broad sense. This structure appears in many fields of mathematics (algebraic topology, algebra, operator theory, combinatorics, Lie theory and algebraic geometry) and mathematical physic. The invention of quantum groups in the eighties stresses the relation with quantum mechanics and other part of physics. The fast development of research of Hopf algebras in recent times has open new areas of interest and important applications in other areas. We mention fusion categories, Nichols algebras, Quantum groups and Yang-Baxter equation, severals generalization of the structure of Hopf algebra (quasi-Hopf algebras, weak Hopf algebras, Hopf algebroids, etc...), 2-categories in Hopf algebras, classification of finite Hopf algebras and aplications in conformal field theory. The aim of this special issue is to collect recent papers concerning these topics.

Prof. Dr. Blas Torrecillas
Prof. Dr. Claudia Menini
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • Hopf Algebras
  • Quantum groups
  • Monoidal Categories
  • Fusion Categories
  • Galois Theories
  • Generalized Hopf Algebra Structures
  • Yang-Baxter equations
  • Categorical methods for Hopf Algebras

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Published Papers (1 paper)

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Research

34 pages, 358 KiB  
Article
Weak Multiplier Hopf Algebras II: Source and Target Algebras
by Alfons Van Daele and Shuanhong Wang
Symmetry 2020, 12(12), 1975; https://doi.org/10.3390/sym12121975 - 30 Nov 2020
Cited by 7 | Viewed by 1794
Abstract
Let (A,Δ) be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct Δ:A⟶M(A⊗A), satisfying certain properties. [...] Read more.
Let (A,Δ) be a weak multiplier Hopf algebra. It is a pair of a non-degenerate algebra A, with or without identity, and a coproduct Δ:A⟶M(A⊗A), satisfying certain properties. In this paper, we continue the study of these objects and construct new examples. A symmetric pair of the source and target maps εs and εt are studied, and their symmetric pair of images, the source algebra and the target algebra εs(A) and εt(A), are also investigated. We show that the canonical idempotent E (which is eventually Δ(1)) belongs to the multiplier algebra M(B⊗C), where (B=εs(A), C=εt(A)) is the symmetric pair of source algebra and target algebra, and also that E is a separability idempotent (as studied). If the weak multiplier Hopf algebra is regular, then also E is a regular separability idempotent. We also see how, for any weak multiplier Hopf algebra (A,Δ), it is possible to make C⊗B (with B and C as above) into a new weak multiplier Hopf algebra. In a sense, it forgets the ’Hopf algebra part’ of the original weak multiplier Hopf algebra and only remembers symmetric pair of the source and target algebras. It is in turn generalized to the case of any symmetric pair of non-degenerate algebras B and C with a separability idempotent E∈M(B⊗C). We get another example using this theory associated to any discrete quantum group. Finally, we also consider the well-known ’quantization’ of the groupoid that comes from an action of a group on a set. All these constructions provide interesting new examples of weak multiplier Hopf algebras (that are not weak Hopf algebras introduced). Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Monoidal Categories)
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