Antipodes and Integrals in Hopf Algebras Over Rings and Quantum Yang-BaxterEquation

摘要

In another paper we showed that every Frobenius algebra over a commutative ringdetermined a solution of the quantum Yang-Baxter equation. Applying this result to Hopf algebras over commutative rings which are finitely generated and projective as modules, we obtain an explicit formula for this solution. It turns out that this solution plays an important role in the structure theory of Hopf algebras over commutative rings. Making use of its properties, we characterize separable Hopf algebras. As a corollary it is shown that the dimension of a simple module over a semisimple Hopf algebra over an algebraically closed field is not divisible by its characteristic. Some results on order of antipode are also obtained.