Quasitriangular and differential structures on bicrossproduct Hopf algebras

1997/01/30 by Edwin Beggs, Shahn Majid, Beggs, E. +1
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Nonlinear Waves and Solitons #Quantum Algebra (math.QA)

paper · pdf · doi:10.48550/arxiv.q-alg/9701041

openalex publication_date 1997/01/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let X=GM be a finite group factorisation. It is shown that the quantum double D(H) of the associated bicrossproduct Hopf algebra H=kM\cobicross k(G) is itself a bicrossproduct kX\cobicross k(Y) associated to a group YX, where Y=G× M^op. This provides a class of bicrossproduct Hopf algebras which are quasitriangular. We also construct a subgroup Y^θX^θ associated to every order-reversing automorphism θ of X. The corresponding Hopf algebra kX^θ\cobicross k(Y^θ) has the same coalgebra as H. Using related results, we classify the first order bicovariant differential calculi on H in terms of orbits in a certain quotient space of X.

Citations

• Differential calculus on compact matrix pseudogroups (quantum groups)

Related

• Finite Group Factorisations and Braiding
• The quantum double for quasitriangular quasi-Hopf algebras
• Classification of differentials and Cartan calculus on bicrossproducts
• Quantum Double for QuasiHopf Algebras
• Quantum differentials on cross product Hopf algebras