In mathematics, a separable algebra is a kind of semisimple algebra. It is a generalization to associative algebras of the notion of a separable field extension.
Definition and first properties
A ring homomorphism (of unital, but not necessarily commutative rings)
is called separable if the multiplication map
admits a section
that is a homomorphism of A-A-bimodules.
If the ring is commmutative and maps into the center of , we call a separable algebra over .
It is useful to describe separability in terms of the element
The reason is that a section σ is determined by this element. The condition that σ is a section of μ is equivalent to
and the condition that σ is a homomorphism of A-A-bimodules is equivalent to the following requirement for any a in A:
Such an element p is called a separability idempotent, since regarded as an element of the algebra it satisfies .
Examples
For any commutative ring R, the (non-commutative) ring of n-by-n matrices is a separable R-algebra. For any , a separability idempotent is given by , where denotes the elementary matrix which is 0 except for the entry in position , which is 1. In particular, this shows that separability idempotents need not be unique.
Separable algebras over a field
A field extension L/K of finite degree is a separable extension if and only if L is separable as an associative K-algebra. If L/K has a primitive element with irreducible polynomial , then a separability idempotent is given by . The tensorands are dual bases for the trace map: if are the distinct K-monomorphisms of L into an algebraic closure of K, the trace mapping Tr of L into K is defined by . The trace map and its dual bases make explicit L as a Frobenius algebra over K.
More generally, separable algebras over a field K can be classified as follows: they are the same as finite products of matrix algebras over finite-dimensional division algebras whose centers are finite-dimensional separable field extensions of the field K. In particular: Every separable algebra is itself finite-dimensional. 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 so that separable K-algebras are finite products of matrix algebras over finite-dimensional division algebras over 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. It can be shown by a generalized theorem of Maschke that an associative K-algebra A is separable if for every field extension the algebra is semisimple.
Group rings
If K is commutative ring 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 .
Equivalent characterizations of separability
There are several equivalent definitions of separable algebras. A K-algebra A is separable if and only if it is projective when considered as a left module of in the usual way. Moreover, an algebra A is separable if and only if it is flat when considered as a right module of in the usual way.
Separable algebras can also be characterized by means of split extensions: A is separable over K if and only if all short exact sequences of A-A-bimodules that are split as A-K-bimodules also split as A-A-bimodules. Indeed, this condition is necessary since the multiplication mapping arising in the definition above is a A-A-bimodule epimorphism, which is split as an A-K-bimodule map by the right inverse mapping given by . The converse can be proven by a judicious use of the separability idempotent (similarly to the proof of Maschke's theorem, applying its components within and without the splitting maps).
Equivalently, the relative Hochschild cohomology groups of in any coefficient bimodule M is zero for . Examples of separable extensions are many including first separable algebras where R is a separable algebra and S = 1 times the ground field. Any ring R with elements a and b satisfying , but ba different from 1, is a separable extension over the subring S generated by 1 and bRa.
Relation to Frobenius 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 particular kind of Frobenius algebra called a symmetric algebra (not to be confused with the symmetric algebra arising as the quotient of the tensor algebra).