Semisimple Algebra - Characterization

Characterization

Let A be a finite dimensional semisimple algebra, and

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

where each

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

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

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

The above result can be restated in a different way. For a semisimple algebra A = A₁ ×...× A_n expressed in terms of its simple factors, consider the units e_i ∈ A_i. The elements E_i = (0,...,e_i,...,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 ≠ j, and Σ 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

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