Separable algebra

A separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.

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Definition and First Properties

Let K be a field. An associative K-algebra A is said to be separable if for every field extension \scriptstyle L/K the algebra \scriptstyle A\otimes_K L 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

p = \sum_{i=1}^n x_i \otimes y_i

in A^e = A \otimes_K A^{\rm op} such that

\sum_{i=1}^n x_i y_i = 1_A

and ap = pa for all a in A. Such an element p is called a separability idempotent, since it satisfies p^2 = p. 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 A^e in the usual way.

Third, an algebra A is separable if and only if it is flat when considered as a right module of A^e 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

e = \sum_{i=1}^n x_i \otimes y_i = \sum_{i=1}^n y_i \otimes x_i

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

Commutative separable algebras

If \scriptstyle L/K is a field extension, then L is separable as an associative K-algebra if and only if the extension of field is separable. If L/K has primitive element a with irreducible polynomial p(x) = (x - a) \sum_{i=0}^{n-1} b_i x^i, then a separability idempotent is given by \sum_{i=0}^{n-1} a^i \otimes_K \frac{b_i}{p'(a)}. The tensorands are dual bases for the trace map: if \sigma_1,\ldots,\sigma_{n} are the distinct K-monomorphisms of L into an algebraic closure of K, the trace mapping Tr of L into K is defined by Tr(x) = \sum_{i=1}^{n} \sigma_i(x). The trace map and its dual bases make explicit L as a Frobenius algebra over K.

Examples

If K is a field and G is a finite group such that the order of G is invertible in K, then the group ring K[G] is a separable K-algebra. A separability idempotent is given by \frac{1}{o(G)} \sum_{g \in G} g \otimes_K g^{-1}.

Separable Extensions for Noncommutative Rings

Let R be an associative ring with unit 1, and S a subring of R containing 1. Notice that an R-R-bimodule (see module theory and homological algebra) restricts to an S-S-bimodule. The ring extension R over S is said to be a separable extension if all short exact sequences of R-R-bimodules that are split as R-S-bimodules also split as R-R-bimodules. For example, the multiplication mapping m : R \otimes_S R \rightarrow R given by m( \sum_i r_i \otimes_S t_i) = \sum_i r_it_i is an R-R-bimodule epimorphism, which is split as an R-S-bimodule epi by the right inverse mapping R \rightarrow R \otimes_S R given by r \mapsto r \otimes 1 . If R is a separable extension over S, then the multiplication mapping is split as an R-R-bimodule epi, so there is a right inverse s of m satisfying for s(1) := e, re = er for all r in R, and m(e) = 1. Conversely, if such an element (called a separability element in the tensor-square) exists, one shows by a judicious use of this element (like Maschke, applying its components within and without the splitting maps) that R is a separable extension of S.

Equivalently, the relative Hochschild cohomology groups H^n(R,S;M) of (R,S) in any coefficient bimodule M is zero for n > 0. Examples of separable extensions are many including first separable algebras where R = separable algebra and S = 1 times the ground field. More interestingly, any ring R with elements a and b satisfying ab = 1, but ba different from 1, is a separable extension over the subring S generated by 1 and bRa.

An interesting theorem in the area is that of J. Cuadra that a separable Hopf-Galois extension R | S has finitely generated natural S-module R. A fundamental fact about a separable extension R | S is that it is left or right semisimple extension: a short exact sequence of left or right R-modules that is split as S-modules, is split as R-modules. In terms of G. Hochschild's relative homological algebra, one says that all R-modules are relative (R,S)-projective. Usually relative properties of subrings or ring extensions, such as the notion of separable extension, serve to promote theorems that say that the over-ring shares a property of the subring. For example, a separable extension R of a semisimple algebra S has R semisimple, which follows from the preceding discussion.

There is the celebrated Jans theorem that a finite group algebra A over a field of characteristic p is of finite representation type if and only if its Sylow p-subgroup is cyclic: the clearest proof is to note this fact for p-groups, then note that the group algebra is a separable extension of its Sylow p-subgroup algebra B as the index is coprime to the characteristic. The separability condition above will imply every finitely generated A-module M is isomorphic to a direct summand in its restricted, induced module. But if B has finite representation type, the restricted module is uniquely a direct sum of multiples of finitely many indecomposables, which induce to a finite number of constituent indecomposable modules of which M is a direct sum. Hence A is of finite representation type if B is. The converse if proven by a similar argument noting that every subgroup algebra B is a B-bimodule direct summand of a group algebra A.

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