Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
1.6
Journals
Axioms
Special Issues
Advances in Hopf Algebras, Tensor Categories and Related Topics
share announcement

Advances in Hopf Algebras, Tensor Categories and Related Topics

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Algebra and Number Theory".

Deadline for manuscript submissions: 31 December 2025 | Viewed by 567

Special Issue Editors

Dr. Tao Yang

E-Mail Website
Guest Editor
College of Science, Nanjing Agricultural University, Nanjing 210095, China
Interests: Hopf algebra; algebraic quantum group; braided tensor category; Yang–Baxter equation
Dr. Florin Felix Nichita

E-Mail Website
Guest Editor
Simion Stoilow Institute of Mathematics of the Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
Interests: (co)algebras; bialgebras; Yang–Baxter equations; Lie (co)algebras; quantum groups; Hopf algebras; duality theories
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Tianshui Ma

E-Mail Website
Guest Editor
School of Mathematics and Statistics, Henan Normal University, Xinxiang 453007, China
Interests: Hopf algebra; Rota–Baxter algebra; Yang–Baxter equation; Yetter–Drinfeld category

Special Issue Information

Dear Colleagues,

The concept of Hopf algebras was first introduced by Heinz Hopf in the 1930s, with the goal of providing a unified framework for the algebraic structures associated with fundamental groups and Lie groups in topology. The definition essentially combines the structures of algebras, rings, and Lie algebras, and is an extension of algebraic structures to express the mathematical properties of Lie groups or Lie algebras. Hopf algebras are not only an important mathematical structure but also a bridge across multiple disciplines. By providing a unified framework, Hopf algebras have allowed mathematicians and physicists to better understand symmetries, quantization, and the complexities of algebraic structures across various fields.

Tensor categories are a powerful and versatile tool and they allow for the study of complex systems by providing a formal framework to describe interactions between objects through tensor products. Their importance spans across many fields, including representation theory, quantum field theory, quantum computing, topology, and more. By enabling the description of symmetries, quantum states, and interactions, tensor categories are a foundational concept in understanding both mathematical structures and physical phenomena.

The relationship between Hopf algebras and tensor categories is deep and significant, particularly in the study of algebraic structures and representation theory. The influence of Hopf algebras on tensor categories is far-reaching. They not only provide concrete mathematical examples for tensor categories but also significantly enrich the theory and practice of tensor categories through applications in quantum groups, representation theory, topological quantum field theory, and beyond. The interaction between Hopf algebras and tensor categories has driven the study of algebraic and physical symmetries, offering crucial tools for understanding complex quantum structures and symmetries across various fields.

Thus, we present this Special Issue of Axioms as a tool to show recent and interesting results in the branches of Hopf algebras, tensor categories, and related topics.

Dr. Tao Yang
Dr. Florin Felix Nichita
Prof. Dr. Tianshui Ma
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. Axioms 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 and their generalizations, such as weak Hopf algebras, (weak) multiplier Hopf algebras, Hopf group-coalgebras, Hom–Hopf algebras, quasi Hopf algebras, Hopf (co)quasigroups, Hopf (co)braces, Hopf algebroids et al.
  • tensor categories, braided monoidal categories, fusion categories, braided crossed categories
  • algebras, coalgebras, symmetry, duality, differential calculi, (co)homologies, groupoids, Yang–Baxter equation
  • (braided) Lie algebras, Lie coalgebras
  • applications of above topics

Benefits of Publishing in a Special Issue

  • Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
  • Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
  • Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
  • External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
  • Reprint: MDPI Books provides the opportunity to republish successful Special Issues in book format, both online and in print.

Further information on MDPI's Special Issue policies can be found here.

Published Papers (1 paper)

Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

42 pages, 509 KB  
Article
Differential Galois Theory and Hopf Algebras for Lie Pseudogroups
by Jean-Francois Pommaret
Axioms 2025, 14(10), 729; https://doi.org/10.3390/axioms14100729 - 26 Sep 2025
Abstract
According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous Spaces” (PHSs) for finite groups [...] Read more.
According to a clever but rarely quoted or acknowledged work of E. Vessiot that won the prize of the Académie des Sciences in 1904, “Differential Galois Theory” (DGT) has mainly to do with the study of “Principal Homogeneous Spaces” (PHSs) for finite groups (classical Galois theory), algebraic groups (Picard–Vessiot theory) and algebraic pseudogroups (Drach–Vessiot theory). The corresponding automorphic differential extensions are such that dimK(L)=∣L/K∣<∞, the transcendence degree trd(L/K)<∞ and trd(L/K)=∞ with difftrd(L/K)<∞, respectively. The purpose of this paper is to mix differential algebra, differential geometry and algebraic geometry to revisit DGT, pointing out the deep confusion between prime differential ideals (defined by J.-F. Ritt in 1930) and maximal ideals that has been spoiling the works of Vessiot, Drach, Kolchin and all followers. In particular, we utilize Hopf algebras to investigate the structure of the algebraic Lie pseudogroups involved, specifically those defined by systems of algebraic OD or PD equations. Many explicit examples are presented for the first time to illustrate these results, particularly through the study of the Hamilton–Jacobi equation in analytical mechanics. This paper also pays tribute to Prof. A. Bialynicki-Birula (BB) on the occasion of his recent death in April 2021 at the age of 90 years old. His main idea has been to notice that an algebraic group G acting on itself is the simplest example of a PHS. If G is connected and defined over a field K, we may introduce the algebraic extension L=K(G); then, there is a Galois correspondence between the intermediate fields K⊂K′⊂L and the subgroups e⊂G′⊂G, provided that K′ is stable under a Lie algebra Δ of invariant derivations of L/K. Our purpose is to extend this result from algebraic groups to algebraic pseudogroups without using group parameters in any way. To the best of the author’s knowledge, algebraic Lie pseudogroups have never been introduced by people dealing with DGT in the spirit of Kolchin; that is, they have only been considered with systems of ordinary differential (OD) equations, but never with systems of partial differential (PD) equations. Full article
(This article belongs to the Special Issue Advances in Hopf Algebras, Tensor Categories and Related Topics)
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-1
Back to TopTop