In
ring theory
In algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies the structure of rings, their r ...
, a branch of mathematics, a semisimple algebra is an
associative artinian algebra over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
which has trivial
Jacobson radical In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition y ...
(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 In mathematics, more specifically ring theory, the Jacobson radical of a ring R is the ideal consisting of those elements in R that annihilate all simple right R-modules. It happens that substituting "left" in place of "right" in the definition y ...
of an algebra over a field is the ideal consisting of all elements that annihilate every simple left-module. The radical contains all
nilpotent ideal In mathematics, more specifically ring theory, an ideal ''I'' of a ring ''R'' is said to be a nilpotent ideal if there exists a natural number ''k'' such that ''I'k'' = 0. By ''I'k'', it is meant the additive subgroup generated by the set of ...
s, 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, ''A
n'' = ''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
:
be a
composition series In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many natura ...
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 ''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
:
By maximality of ''J''
1 as an ideal in ''J''
2 and also the semisimplicity of ''A'', the algebra
:
is simple. Proceed by induction in similar fashion proves the claim. For example, ''J''
3 is the Cartesian product of simple algebras
:
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 element
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
s 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
Joseph Henry Maclagan Wedderburn FRSE FRS (2 February 1882 – 9 October 1948) was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a fi ...
completely classifies finite-dimensional semisimple algebras over a field
. Any such algebra is isomorphic to a finite product
where the
are natural numbers, the
are
division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible.
Definitions
Formally, we start with a non-zero algebra ''D'' over a fie ...
s over
, and
is the algebra of
matrices over
. 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