The structure of Frobenius algebras and separable algebras

Abstract

We present a unified apoach to the study of separable and Frobenius algebras. The crucial observation is thsat both cases are related to the nonlinear equation $R^{12}{R}^{23}=R^{23}{R}^{13}=R^{13}{R}^{12}$, called the FS-equation. Given a solution of the FS-equation satisfying certain normalizing conditions, we can construct a Frobenius algebra or a separable algebra A(R). The main result of this paper in the structure of this two fundamental types of algebras: a finitely generated projective Frobenius or separable $k$-algebra $A$ is isomorphic to such A(R). If $A$ is free as a $k$-module, then A(R) can be described using generators and relations. A new characterisation of Frobenius extensions is given: $B\subset A$ is Frobenius if and only if $A$ has a $B$-coring structure such that the comultiplication $\Delta$ is an $A$-bimodule map.

arXiv comment34 pages

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