Representations of the quantum double of quasitriangular Hopf algebras

Miriam Cohen, Sara Westreich

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Conjugacy classes for groups were generalized in previous works of the authors to semisimple Hopf algebras over algebraically closed fields of characteristic zero. An essential property of the set of conjugacy classes {Cj} is that they are irreducible representations of the quantum double D(H) of H. We show that more is true. When (H, R) is quasitriangular, then any irreducible representations of D(H) is a direct summand of a tensor product of the form of {Cj} Vi, where Vi is an irreducible representation of H. The proof follows by applying a partition defined in our 2011 paper on the indices of the irreducible characters of D(H) and identifying the partition with the equivalence classes obtained from Nichols-Richmond equivalence relations on the set of the simple subcoalgebra of D(H). For any block of this partition, there is a unique Cj whose character as a D(H)-representation belongs to it. When V is a 1-dimensional representation of H then Cj ⊗ V is an irreducible representation of D(H), isomorphic to Cj as a vector space endowed with a twisted action of D(H). We apply the above results to H=kG, where G are the groups S3 and D4, and decompose all Cj ⊗ Vi into their irreducible components.

Original languageEnglish
Title of host publicationAmitsur Centennial Symposium, 2021
EditorsAvinoam Mann, Louis H. Rowen, David J. Saltman, Aner Shalev, Lance W. Small, Uzi Vishne
PublisherAmerican Mathematical Society
Number of pages16
ISBN (Print)9781470475550
StatePublished - 2024
EventAmitsur Centennial Symposium, 2021 - Jerusalem, Israel
Duration: 1 Nov 20214 Nov 2021

Publication series

NameContemporary Mathematics
ISSN (Print)0271-4132
ISSN (Electronic)1098-3627


ConferenceAmitsur Centennial Symposium, 2021

Bibliographical note

Publisher Copyright:
© 2024 Miriam Cohen and Sara Westreich.


  • Hopf conjugacy classes
  • Quantum double; Drinfrld double
  • Quasitriangular Hopf algebras


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