  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 $GL_n(k)
\times M_n(k)\otimes M_n(k) \to M_n(k) \otimes M_nk$ $(u, R) \to (u\otimes u)R
(u\otimes u)^{-1}$, the representatives of wich are invertible solutions of
length n of the pentagon equation.
