Polynomial invariants for a semisimple and cosemisimple Hopf algebra of finite dimension

Abstract

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra A over a field k by using the braiding structures of A . The coefficients of polynomial invariants are integers if k is a finite Galois extension of Q , and A is a scalar extension of some finite-dimensional semisimple Hopf algebra over Q . Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of A , and recognize the difference between the representation category and the representation ring of A . Actually, by computing and comparing polynomial invariants, we find new examples of pairs of Hopf algebras whose representation rings are isomorphic, but whose representation categories are distinct.