Separable Algebra - Definition and First Properties

Definition and First Properties

Let K be a field. An associative K-algebra A is said to be separable if for every field extension the algebra is semisimple.

There is a classification theorem for separable algebras: separable algebras are the same as finite products of matrix algebras over division algebras whose centers are finite dimensional separable field extensions of the field K. If K is a perfect field --- for example a field of characteristic zero, or a finite field, or an algebraically closed field --- then every extension of K is separable. As a result, if K is a perfect field, separable algebras are the same as finite products of matrix algebras over division algebras whose centers are finite-dimensional field extensions of the field K. In other words, if K is a perfect field, there is no difference between a separable algebra over K and a finite-dimensional semisimple algebra over K.

There are a several equivalent characterizations of separable algebras. First, an algebra A is separable if and only if there exists an element

in such that

and ap = pa for all a in A. Such an element p is called a separability idempotent, since it satisfies . A generalized theorem of Maschke shows these this characterization of separable algebras is equivalent to the definition given above.

Second, an algebra A is separable if and only if it is projective when considered as a left module of in the usual way.

Third, an algebra A is separable if and only if it is flat when considered as a right module of in the usual (but perhaps not quite standard) way. See Aguiar's note below for more details.

Furthermore, a result of Eilenberg and Nakayama that any separable algebra can be given the structure of a symmetric Frobenius algebra. Since the underlying vector space of a Frobenius algebra is isomorphic to its dual, any Frobenius algebra is necessarily finite dimensional, and so the same is true for separable algebras.

A separable algebra is said to be strongly separable if there exists a separability idempotent that is symmetric, meaning

An algebra is strongly separable if and only if its trace form is nondegenerate, thus making the algebra into a special Frobenius algebra.