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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 ...
Weyl's theorem implies (in fact is equivalent to) that the enveloping algebra of a finite-dimensional representation is a semisimple ring in the following way.. Given a finite-dimensional Lie algebra representation : (), let be the associative subalgebra of the endomorphism algebra of V generated by ().
A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible) if it is the direct sum of simple (irreducible) submodules. For a module M, the following are equivalent: M is semisimple; i.e., a direct sum of irreducible modules. M is the sum of its irreducible submodules.
A representation is called semisimple or completely reducible if it can be written as a direct sum of irreducible representations. This is analogous to the corresponding definition for a semisimple algebra.
Now Maschke's theorem says that any finite-dimensional representation of a finite group is a direct sum of simple representations (provided the characteristic of the base field does not divide the order of the group). So in the case of finite groups with this condition, every finite-dimensional representation is semi-simple.
More formally, a "representation" means a homomorphism from the group to the automorphism group of an object. If the object is a vector space we have a linear representation. Some people use realization for the general notion and reserve the term representation for the special case of linear representations. The bulk of this article describes ...
In mathematics, a Lie algebra is reductive if its adjoint representation is completely reducible, hence the name. More concretely, a Lie algebra is reductive if it is a direct sum of a semisimple Lie algebra and an abelian Lie algebra : g = s ⊕ a ; {\displaystyle {\mathfrak {g}}={\mathfrak {s}}\oplus {\mathfrak {a}};} there are alternative ...
In case of spin 1/2 particles, it is possible to find a construction that includes both a finite-dimensional representation and a scalar product preserved by this representation by associating a 4-component Dirac spinor with each particle.