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 ''A''² = ≠ . 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 ''A''² is an ideal of ''A'' and ''A'' is simple, ''A''² = ''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 ''e''₁ is the multiplicative identity in ''A''₁ (all simple algebras possess a multiplicative identity), then (''a''₁, ''a''₂, ...) · (''e''₁, 0, ...) = (''a''₁, 0..., 0) lies in some nilpotent ideal of Π ''A_i''. This implies, for all ''b'' in ''A''₁, ''a''₁''b'' is nilpotent in ''A''₁, i.e. ''a''₁ ∈ 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

:$\ = J_0 \subset \cdots \subset J_n \subset A$

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

:$A \simeq J_1 \times J_2/J_1 \times J_3/J_2 \times ... \times J_n/ J_ \times A / J_n$

where each

:$J_/J_i \,$

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 ×...× ''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

#''E_iE_j'' = 0 for ''i'' ≠ ''j'' (such a set of idempotents is called '' central orthogonal''),
#Σ ''E_i'' = 1,
#''A'' is isomorphic to the Cartesian product of simple algebras ''E''₁ ''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 $\prod M_(D_i)$ where the $n_i$ are natural numbers, the $D_i$ are division algebras over $k$, and $M_(D_i)$ 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 Artin-Wedderburn theorem.

References

Springer Encyclopedia of Mathematics

{{DEFAULTSORT:Semisimple Algebra
Algebras