Semisimple algebra

In ring theory, a branch of mathematics, a semisimple algebra is an associative artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in the Jacobson radical). If the algebra is finite-dimensional this is equivalent to saying that it can be expressed as a Cartesian product of simple subalgebras.

Definition
The Jacobson radical of an algebra over a field is the ideal consisting of all elements that annihilate every simple left-module. The radical contains all nilpotent ideals, and if the algebra is finite-dimensional, the radical itself is a nilpotent ideal. A finite-dimensional algebra is then said to be semisimple if its radical contains only the zero element.

An algebra A is called simple if it has no proper ideals and A2 = {ab | a, b ∈ A} ≠ {0}. As the terminology suggests, simple algebras are semisimple. The only possible ideals of a simple algebra A are A and {0}. Thus if A is simple, then A is not nilpotent. Because A2 is an ideal of A and A is simple, A2 = A. By induction, An = A for every positive integer n, i.e. A is not nilpotent.

Any self-adjoint subalgebra A of n × n matrices with complex entries is semisimple. Let Rad(A) be the radical of A. Suppose a matrix M is in Rad(A). Then M*M lies in some nilpotent ideals of A, therefore (M*M)k = 0 for some positive integer k. By positive-semidefiniteness of M*M, this implies M*M = 0. So M x is the zero vector for all x, i.e. M = 0.

If {Ai} is a finite collection of simple algebras, then their Cartesian product A=Π Ai is semisimple. If (ai) is an element of Rad(A) and e1 is the multiplicative identity in A1 (all simple algebras possess a multiplicative identity), then (a1, a2, ...) · (e1, 0, ...) = (a1, 0..., 0) lies in some nilpotent ideal of Π Ai. This implies, for all b in A1, a1b is nilpotent in A1, i.e. a1 ∈ Rad(A1). So a1 = 0. Similarly, ai = 0 for all other i.

It is less apparent from the definition that the converse of the above is also true, that is, any finite-dimensional semisimple algebra is isomorphic to a Cartesian product of a finite number of simple algebras.

Characterization
Let A be a finite-dimensional semisimple algebra, and


 * $$\{0\} = J_0 \subset \cdots \subset J_n \subset A$$

be a composition series of A, then A is isomorphic to the following Cartesian product:


 * $$A \simeq J_1 \times J_2/J_1 \times J_3/J_2 \times ... \times J_n/ J_{n-1} \times A / J_n $$

where each


 * $$J_{i+1}/J_i \,$$

is a simple algebra.

The proof can be sketched as follows. First, invoking the assumption that A is semisimple, one can show that the J1 is a simple algebra (therefore unital). So J1 is a unital subalgebra and an ideal of J2. Therefore, one can decompose


 * $$J_2 \simeq J_1 \times J_2/J_1 .$$

By maximality of J1 as an ideal in J2 and also the semisimplicity of A, the algebra


 * $$J_2/J_1 \,$$

is simple. Proceed by induction in similar fashion proves the claim. For example, J3 is the Cartesian product of simple algebras


 * $$J_3 \simeq J_2 \times J_3 / J_2 \simeq J_1 \times J_2/J_1 \times J_3 / J_2.$$

The above result can be restated in a different way. For a semisimple algebra A = A1 ×...× An expressed in terms of its simple factors, consider the units ei ∈ Ai. The elements Ei = (0,...,ei,...,0) are idempotent elements in A and they lie in the center of A. Furthermore, Ei A = Ai, EiEj = 0 for i ≠ j, and Σ Ei = 1, the multiplicative identity in A.

Therefore, for every semisimple algebra A, there exists idempotents {Ei} in the center of A, such that


 * 1) EiEj = 0 for i ≠ j (such a set of idempotents is called central orthogonal),
 * 2) Σ Ei = 1,
 * 3) A is isomorphic to the Cartesian product of simple algebras E1 A ×...× En A.

Classification
A theorem due to Joseph Wedderburn completely classifies finite-dimensional semisimple algebras over a field $$ k$$. Any such algebra is isomorphic to a finite product $$ \prod M_{n_i}(D_i) $$ where the $$ n_i $$ are natural numbers, the $$ D_i $$ are division algebras over $$k $$, and $$ M_{n_i}(D_i) $$ is the algebra of $$ n_i \times n_i $$ matrices over $$ D_i$$. This product is unique up to permutation of the factors.

This theorem was later generalized by Emil Artin to semisimple rings. This more general result is called the Wedderburn–Artin theorem.