Biproduct Quasi-Hopf Algebras of Rank 2

2025/08/01 by Daniel Bulacu, Bulacu, Daniel, Matteo Misurati +1
Mathematics · #16W30 #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2508.00597

openalex publication_date 2025/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Inspired by the work of Radford, for H an arbitrary quasi-Hopf algebra we describe all the Hopf algebras of dimension 2 within the braided category of left Yetter-Drinfeld modules over H and determine the biproduct quasi-Hopf algebras defined by them. Classes of such biproduct quasi-Hopf algebras are obtained by taking H as the Hopf algebra of functions on a group G, endowed with the quasi-Hopf algebra structure provided by a non-trivial 3-cocycle on G (especially when G is a finite cyclic group or the double dihedral group), or as being a quasi-Hopf algebra with radical of codimension two. In this way we uncover new classes of basic quasi-Hopf algebras of even dimension, as well as new classes of tensor categories.

Citations

• Quasi-Hopf Algebras of Dimension 6
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• Diagonal Crossed Products by Duals of Quasi-Quantum Groups
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Related

• Yetter-Drinfeld categories for quasi-Hopf algebras
• Basic quasi-Hopf algebras of dimension n³
• More properties of Yetter-Drinfeld category over dual quasi-Hopf algebras
• Doi-Hopf modules and Yetter-Drinfeld modules for quasi-Hopf algebras
• Structure theorems for bicomodule algebras over quasi-Hopf algebras, weak Hopf algebras and braided Hopf algebras