Advances in Hopf Algebras, Tensor Categories and Related Topics

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

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• 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