Semisimple algebra explained

See also: Semisimple module. 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 = ≠ . As the terminology suggests, simple algebras are semisimple. The only possible ideals of a simple algebra A are A and . 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 is a finite collection of simple algebras, then their Cartesian product A=Π 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 Π 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.

Characterization

Let A be a finite-dimensional semisimple algebra, and

\{ 0 \} = J _{0} \subset … \subset J _{n} \subset A

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

A \simeq J _{1} x J _{2/J} _{1} x J _{3/J} _{2} x ... x J _{n/} J _{n-1} x 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} x 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} x J _{3} / J _{2} \simeq J _{1} x J _{2/J} _{1} x J _{3} / J _{2.}

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 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.

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} x 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

Springer Encyclopedia of Mathematics

Notes and References

1. Book: Anthony Knapp. Advanced Algebra, Chap. II: Wedderburn-Artin Ring Theory. 2007. Springer Verlag.