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

We introduce new polynomial invariants of a finite-dimensional semisimple and cosemisimple Hopf algebra AA over a field k by using the braiding structures of AA. The coefficients of polynomial invariants are integers if k is a finite Galois extension of QQ, and AA is a scalar extension of some finite-dimensional semisimple Hopf algebra over QQ. Furthermore, we show that our polynomial invariants are indeed tensor invariants of the representation category of AA, and recognize the difference between the representation category and the representation ring of AA. 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.