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 RR = RR = RR, called the FS-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 finitely generated projective Frobenius or separable k-algebra A is isomorphic to such an A(R). If A is a free k-algebra, then A(R) can be described using generators and relations. A new characterization of Frobenius extensions is given: B ⊂ A is Frobenius if and only if A has a B-coring structure (A,∆, ε) such that the comultiplication ∆ : A→ A⊗B A is an A-bimodule map. 0 Introduction Frobenius extensions in noncommutative ring theory have been introduced by Kasch [32], as generalizations of the classical notion of Frobenius algebras over a field (see also [41], [42], [45]). The notion of Frobenius extension has turned out to be a fundamental one, and many generalizations have appeared in the literature. Roughly stated, an object O satisfies a “Frobenius-type” property if two conditions hold: a “finiteness” condition (for example finite dimensional) and a “symmetry” condition (for example O has an isomorphic dualO∗). O can be an algebra ([18]), a coalgebra ([19]), a Lie algebra ([21]), a Hopf algebra ([44]) or, in the most general case, a functor between two categories ([15]). In fact an object O over a field k is Frobenius if the forgetful functor F : ORep → kM from the category of representations of O to vector spaces is Frobenius, and this means that F has a left adjoint which is also a right adjoint. In recent years, many important new results about “Frobenius objects” came about (see [2]-[5], [7]-[10], [12], [15], [25], [26], [30]). Let us mention a few of them, illustrating the importance of the concept. The homology of a compact oriented manifold is a Frobenius algebra. In [2] (see also [5]), ∗Research supported by the project “Hopf algebras and co-Galois theory” of MCT of Romanian government.