Search results
Results from the WOW.Com Content Network
The irreducible complex characters of a finite group form a character table which encodes much useful information about the group G in a concise form. Each row is labelled by an irreducible character and the entries in the row are the values of that character on any representative of the respective conjugacy class of G (because characters are class functions).
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 ...
Irreducible representations are always indecomposable (i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangular unipotent matrices is indecomposable but reducible.
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 ().
In mathematics, especially in the area of abstract algebra known as module theory, a semisimple module or completely reducible module is a type of module that can be understood easily from its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring .
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 ...
The identity component of PGL(2,R) (sometimes called PSL(2,R)) is a real reductive group that cannot be viewed as an algebraic group. Similarly, SL(2) is simply connected as an algebraic group over any field, but the Lie group SL(2,R) has fundamental group isomorphic to the integers Z, and so SL(2,R) has nontrivial covering spaces.
In representation theory, a branch of mathematics, the theorem of the highest weight classifies the irreducible representations of a complex semisimple Lie algebra. [ 1 ] [ 2 ] There is a closely related theorem classifying the irreducible representations of a connected compact Lie group K {\displaystyle K} . [ 3 ]