The structure of Frobenius algebras and separable algebras

We present a unified approach to the study of separable and Frobenius algebras. The crucial observation is that both types of algebras are related to the nonlinear equation (RR23)-R-12=(RR13)-R-23=(RR12)-R-13, called the FS-equation. Solutions of the FS-equation automatically satisfy the braid equation, an equation that is in a sense equivalent to the quantum Yang-Baxter equation. Given a solution to the FS-equation satisfying a certain normalizing condition, we can construct a Frobenius algebra or a separable algebra A(R) - the normalizing condition is different in both cases. The main result of this paper is the structure of these two fundamental types of algebras: a finite dimensional Frobenius or separable k-algebra A is isomorphic to such an A(R). A(R) can be described using generators and relations. A new characterization of Frobenius extensions is given: B subset of A is Frobenius if and only if A has a B-coring structure (A, Delta, epsilon) such that the comultiplication Delta: A --> Ax(B) A is an A-imodule map.