On antipodes and integrals in Hopf algebras over rings and the quantum Yang-Baxter equation

The authors showed previously (on Frobenius algebras and quantum Yang-Baxter equation, II, preprint, TRITA-MAT-1995, February 1995) that every Frobenius algebra over a commutative ring defines 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 can be expressed in terms of the integral and antipode. We use this solution to characterize separable Hopf algebras over rings. Some results on the order of the antipode are also obtained. (C) 1997 Academic Press.