Heisenberg duoble, pentagon equation, structure and classification of finite dimensional Hopf algebras

Abstract

The study of the pentagon (fusion) equation leds to the Structure and the Classification theorem for finite dimenasional Hopf algebras: there exists a one to one correspondence between the set of types of n-dimensional Hopf algebtras and the set of the orbits of the resticted Jordan action $G{L}_n(k)\times M_n(k)\otimes M_n(k)\to M_n(k)\otimes M_{n}k$ $(u,R)\to (u\otimes u)R(u\otimes u{)}^{-1}$, the representatives of wich are invertible solutions of length n of the pentagon equation.

22 pg, latex

References in corpus (6)

• Heisenberg Double and Pentagon Relation
• Quantum Random Walks and Time Reversal
• The Hopf modules category and the Hopf equation
• Separable functors for Doi-Hopf modules. Applications
• Pentagon equation and matrix bialgebras
• On pointed Hopf algebras and Kaplansky's 10th conjecture

Cited by in corpus (8)

• Simplex and Polygon Equations
• The pentagon relation and incidence geometry
• Set-theoretic solutions of the Pentagon Equation
• Heisenberg double H(B^*) as a braided commutative Yetter-Drinfeld module algebra over the Drinfeld double
• A Heisenberg double addition to the logarithmic Kazhdan--Lusztig duality
• Set-theoretical solutions of the pentagon equation on Clifford semigroups
• On the Structure of Set-Theoretic Polygon Equations
• Set-theoretical solutions to the pentagon equation: a survey