Heisenberg Double, Pentagon Equation, Structure and Classification of Finite‐Dimensional Hopf Algebras

Abstract

The study of the pentagon equation leads to results on the structure and classification of finite quantum groups. It is proved that L is a finite‐dimensional Hopf algebra if and only if there exists an invertible matrix R, solution of the pentagon equation R¹²R¹³R²³=R²³R¹², such that L≅ P(n, R); the Hopf algebra structure of P(n, R) is explicitly described using generators and relations. Finally, it is proved that there exists a one‐to‐one correspondence between the set of types of n‐dimensional Hopf algebras and the set of orbits of the action GL_n(k)×(M_n(k)⊗M_n(k))→ M_n(k)⊗ M_n(k),(u, R)→(u⊗u)R(u⊗u)⁻¹, the representatives of which are invertible solutions of length n for the pentagon equation.