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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 ().
These groups are characterized by i) an n-fold proper rotation axis C n; ii) n 2-fold proper rotation axes C 2 normal to C n; iii) a mirror plane σ h normal to C n and containing the C 2 s. The D 1h group is the same as the C 2v group in the pyramidal groups section. The D 8h table reflects the 2007 discovery of errors in older references. [4]
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.
Every finite-dimensional representation of sl(2,C) decomposes as a direct sum of irreducible representations. This claim follows from the general result on complete reducibility of semisimple Lie algebras, [ 11 ] or from the fact that sl(2, C ) is the complexification of the Lie algebra of the simply connected compact group SU(2). [ 12 ]
The most basic example is the Lie algebra of matrices with the commutator as Lie bracket, or more abstractly as the endomorphism algebra of an n-dimensional vector space, (). This is the Lie algebra of the general linear group GL( n ), and is reductive as it decomposes as g l n = s l n ⊕ k , {\displaystyle {\mathfrak {gl}}_{n}={\mathfrak {sl ...
For n = 3 the obvious analogue of the (n − 1)-dimensional representation is reducible – the permutation representation coincides with the regular representation, and thus breaks up into the three one-dimensional representations, as A 3 ≅ C 3 is abelian; see the discrete Fourier transform for representation theory of cyclic groups.
The notion of a reductive dual pair makes sense over any field F, which we assume to be fixed throughout.Thus W is a symplectic vector space over F.. If W 1 and W 2 are two symplectic vector spaces and (G 1, G′ 1), (G 2, G′ 2) are two reductive dual pairs in the corresponding symplectic groups, then we may form a new symplectic vector space W = W 1 ⊕ W 2 and a pair of groups G = G 1 × G ...