Semisimple algebra

Short description: Associative Artinian algebra with a trivial Jacobson radical

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 $A^2=\{ab|a,b\in A\}\ne \{0\}$. 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 $\mathrm{Rad}(A)$ be the radical of $A$. Suppose a matrix $M$ is in $\mathrm{Rad}(A)$. 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 $\mathrm{Rad}(A)$ 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 $A_1$, $a_{1}b$ is nilpotent in $A_1$, i.e. $a_1\in \mathrm{Rad}(A_1)$. 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, $E_{i}A=A_i,E_i{E}_j=0$ 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

1. $E_i{E}_j=0$ for $i\ne j$ (such a set of idempotents is called central orthogonal),
2. $\sum E_i=1$,
3. $A$ is isomorphic to the Cartesian product of simple algebras $E_{1}A\times \dots \times E_{n}A$.

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 $n_i\times n_i$ 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 Wedderburn–Artin theorem.

References

Springer Encyclopedia of Mathematics