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Concept
Semisimple representation
Summary
In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematical notion of semisimplicity.
Many representations that appear in applications of representation theory are semisimple or can be approximated by semisimple representations. A semisimple module over an algebra over a field is an example of a semisimple representation. Conversely, a semisimple representation of a group G over a field k is a semisimple module over the group ring k[G].
Let V be a representation of a group G; or more generally, let V be a vector space with a set of linear endomorphisms acting on it. In general, a vector space acted on by a set of linear endomorphisms is said to be simple (or irreducible) if the only invariant subspaces for those operators are zero and the vector space itself; a semisimple representation then is a direct sum of simple representations in that sense.
The following are equivalent:
V is semisimple as a representation.
V is a sum of simple subrepresentations.
Each subrepresentation W of V admits a complementary representation: a subrepresentation W such that .
The equivalences of the above conditions can be shown based on the next lemma, which is of independent interest:
Proof of the lemma: Write where are simple representations. Without loss of generality, we can assume are subrepresentations; i.e., we can assume the direct sum is internal. Now, consider the family of all possible direct sums with various subsets . Put the partial ordering on it by saying the direct sum over K is less than the direct sum over J if . By Zorn's lemma, we can find a maximal such that . We claim that . By definition, so we only need to show that . If is a proper subrepresentatiom of then there exists such that . Since is simple (irreducible), . This contradicts the maximality of , so as claimed.
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