Semisimple Algebra - Definition

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 not nilpotent, then A is semisimple. 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 {A_i} is a finite collection of simple algebras, then their Cartesian product ∏ A_i is semisimple. If (a_i) 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 ∏ A_i. This implies, for all b in A₁, a₁b is nilpotent in A₁, i.e. a₁ ∈ Rad(A₁). So a₁ = 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. The following is a semisimple algebra that appears not to be of this form. Let A be an algebra with Rad(A) ≠ A. The quotient algebra B = A ⁄ Rad(A) is semisimple: If J is a nonzero nilpotent ideal in B, then its preimage under the natural projection map is a nilpotent ideal in A which is strictly larger than Rad(A), a contradiction.