Central simple algebras

In this appendix, we will review some well known facts about central simple algebras (see [Bou] Alg. Ch. 8, [Re] and [We 2]).

Let K be a commutative field. A K-algebraA is a ring (associative and with a unit element) endowed with a injective ring homomorphism of K into the center of A which maps 1 to 1. The K-algebra A is said to be central if the image of K in A is exactly the center of A. The K-algebra A is said to be simple if any right A-module is semi-simple and if, up to isomorphisms, there is only one simple right A-module.

We will consider only simple K-algebras which are of finite dimension as K-vector spaces. If A is such a simple K-algebra, one denotes by [A : K] its dimension over K. If we assume moreover that A is central then one has [A : K] = d2 for some positive integer d.

If V is “the” unique simple right A-module for some central simple K-algebra A as before, D = HomA(V, V) is a central division algebra overK (i.e. a skew field with center K) and A is canonically isomorphic to HomD(V, V). In particular, if we choose a basis of the finite dimensional right D-vector space V, we get an isomorphism of A with the matrix algebra glr(D)(r = dimD(V)).