Quasitriangular structures on abelian extensions of ℤ₂

2021/08/01 by Kun Zhou, Zhou, Kun
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA) #Rings and Algebras (math.RA)

paper · pdf · doi:10.48550/arxiv.2108.00446

openalex publication_date 2021/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The aim of this paper is to give all quasitriangular structures on a class of semisimple Hopf algebras constructed through abelian extensions of \Bbbkℤ₂ by \Bbbk^G for an abelian group G. We first introduce the concept of symmetry of quasitriangular structures of Hopf algebras and obtain some related propositions which can be used to simplify our calculations of quasitriangular structures. Secondly, we find that quasitriangular structures of these semisimple Hopf algebras can do division-like operations. Using such operations we transform the problem of solving the quasitriangular structures into solving general solutions and giving a special solution. Then we give all general solutions and get a necessary and sufficient condition for the existence of a special solution.

Citations

• Invariants of 3-manifolds via link polynomials and quantum groups

Related

• On the quasitriangular structures of abelian extensions of Z₂
• On semisimple quasitriangular Hopf algebras of dimension dqⁿ
• On quasitriangular Hopf algebras related to the Borel subalgebra of sl(2)
• Quasitriangular structures on cocommutative Hopf algebras
• On quasitriangular structures in Hopf algebras arising from exact group\n factorizations