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 A^2=\{ab|a,b\in A\}\ne\{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 A^2 is an ideal of A and A is simple, A^2=A. By induction, A^n=A for every positive integer n, i.e. A is not nilpotent.
Any self-adjoint subalgebra A of n\times n matrices with complex entries is semisimple. Let \operatorname{Rad}(A) be the radical of A. Suppose a matrix M is in \operatorname{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 Mx is the zero vector for all x, i.e. M=0.
If {A_i} is a finite collection of simple algebras, then their Cartesian product A=\prod A_i is semisimple. If (A_i) is an element of \operatorname{Rad}(A) and e_1 is the multiplicative identity in A_1 (all simple algebras possess a multiplicative identity), then (a_1,a_2,\dots)\cdot(e_1,0,\dots)=(a_1,0,\dots) lies in some nilpotent ideal of \prod A_i. This implies, for all b in A_1, a_1b is nilpotent in A_1, i.e. a_1\in\operatorname{Rad}(A_1). So a_1=0. Similarly, a_i=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 J_1 is a simple algebra (therefore unital). So J_1 is a unital subalgebra and an ideal of J_2. Therefore, one can decompose
J_2 \simeq J_1 \times J_2/J_1 .
By maximality of J_1 as an ideal in J_2 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, J_3 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=A_1\times\cdots\times A_n expressed in terms of its simple factors, consider the units e_i\in A_i. The elements E_i=(0,\dots,e_i,\dots,0) are idempotent elements in A and they lie in the center of A Furthermore, E_i A=A_i,E_iE_j=0 for i \ne j, and \sum E_i=1, the multiplicative identity in A.
Therefore, for every semisimple algebra A, there exists idempotents \{E_i\} in the center of A, such that
E_iE_j=0fori\ne j(such a set of idempotents is called central orthogonal),\sum E_i=1,Ais isomorphic to the Cartesian product of simple algebrasE_1A\times\dots\times E_nA.
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.[1]
This theorem was later generalized by Emil Artin to semisimple rings. This more general result is called the Wedderburn–Artin theorem.
References
- ^ Anthony Knapp (2007). Advanced Algebra, Chap. II: Wedderburn-Artin Ring Theory. Springer Verlag.