Algebra Seminar

Let A be a Frobenius algebra over a commutative ring R. It turns out that Frobenius structure on A defines canonically a solution of the Yang-Baxter equation. Using the fact that finite projective Hopf algebras over rings with trivial Picard group are Frobenius, we obtain an explicit formula for this solution in terms of the integral and antipode. We use this formula to prove that the antipode S in finite projective Hopf algebras has a finite order. Further, we prove that if the algebra has a constant rank N, as a projective module, then S^4N=id.