Separable algebra

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.

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. p := \sigma(1) = \sum a_i \otimes b_i \in A \otimes_K A The reason is that a section σ is determined by this element. The condition that σ is a section of μ is equivalent to \sum a_i b_i = 1 and the condition that σ is a homomorphism of A-A-bimodules is equivalent to the following requirement for any a in A: \sum a a_i \otimes b_i = \sum a_i \otimes b_i a. Such an element p is called a separability idempotent, since regarded as an element of the algebra A \otimes A^{\rm op} it satisfies p^2 = p . For any commutative ring R, the (non-commutative) ring of n-by-n matrices M_n(R) is a separable R-algebra. For any 1 \le j \le n , a separability idempotent is given by \sum_{i=1}^n e_{ij} \otimes e_{ji} , where e_{ij} denotes the elementary matrix which is 0 except for the entry in the entry, which is 1. In particular, this shows that separability idempotents need not be unique. Indeed, they are not even unique up to automorphism. 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 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. 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 L/K the algebra A\otimes_K L is semisimple. If K is commutative ring and G is a finite group such that the order of G is invertible in K, then the group algebra K[G] is a separable K-algebra. A separability idempotent is given by \frac{1}{o(G)} \sum_{g \in G} g \otimes g^{-1} . 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 \mu : A \otimes_K A \rightarrow A 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 A \rightarrow A \otimes_K A given by a \mapsto a \otimes 1 . 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 H^n(R,S;M) 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. 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 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). If K is commutative, A is a finitely generated projective separable K-module, then A is a symmetric Frobenius algebra. Samuel Eilenberg and Tadasi Nakayama, On the dimension of modules and algebras. II. Frobenius algebras and quasi-Frobenius rings, Nagoya Math. J. Volume 9 (1955), 1–16.

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