specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group...

said to be reductive if the adjoint representation is semisimple. Certainly, every (finite-dimensional) semisimple Lie algebra g {\displaystyle {\mathfrak...

representations of semisimple Lie algebras are completely understood, after work of Élie Cartan. A representation of a semisimple Lie algebra 𝖌 is analysed...

of the same notion, see Semisimple representation. A module over a (not necessarily commutative) ring is said to be semisimple (or completely reducible)...

In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any...

In mathematics, the representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras...

connected group G {\displaystyle G} admitting a faithful semisimple representation which remains semisimple over its algebraic closure k a l {\displaystyle k^{al}}...

linear operator on V, then V is a semisimple representation of T. Equivalently, a linear operator is semisimple if its minimal polynomial is a product of...

complete reducibility says a finite-dimensional representation of a semisimple compact Lie group is semisimple. A square matrix (in other words a linear operator...

In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp...

construct any general representation theory, but specific special cases have been studied, sometimes using ad hoc techniques. The semisimple Lie groups have...

classical Lie group is a fundamental representation. Any finite-dimensional irreducible representation of a semisimple Lie group or Lie algebra can be constructed...

representation theory of a Lie group that is neither a compact group nor a semisimple group. It is fundamental in theoretical physics. In a physical theory...

the elements of S. Equivalently, for any set S of mutually commuting semisimple linear transformations of a finite-dimensional vector space V there exists...

If in addition g {\displaystyle {\mathfrak {g}}} is semisimple, then the adjoint representation presents g {\displaystyle {\mathfrak {g}}} as a linear...

certain types of Lie groups, namely compact and semisimple groups, every finite-dimensional representation decomposes as a direct sum of irreducible representations...

adjoint representation of G: Int ⁡ ( g ) = Ad ⁡ ( G ) {\displaystyle \operatorname {Int} ({\mathfrak {g}})=\operatorname {Ad} (G)} . If G is semisimple, the...

coadjoint representation is the dual representation of an adjoint representation. complete “completely reducible" is another term for "semisimple". complex...

can associate a semisimple character (corresponding to some semisimple element s of the dual group), and a unipotent representation of the centralizer...

of K does divide |G|, is harder mainly because with K[G] not semisimple, a representation can fail to be irreducible without splitting as a direct sum...

representations (specifically in the representation theory of semisimple Lie algebras). Let g {\displaystyle {\mathfrak {g}}} be a semisimple Lie algebra over a field...

is semisimple, then every element of g {\displaystyle {\mathfrak {g}}} is a linear combination of commutators, in which case every representation of g...

non-rational, but are still semisimple representations. A further class of transformations come from the logarithmic representation of the general linear group...

of G is divisible by the characteristic of K, the group algebra is not semisimple, hence has non-zero Jacobson radical. In that case, there are finite-dimensional...

characteristic zero, every finite-dimensional representation of a semisimple Lie algebra is semisimple (that is, a direct sum of irreducible representations)...

happens in the representation theory of compact connected Lie groups, or the finite-dimensional representation theory of complex semisimple Lie algebras...

}(e^{X})} . Suppose now that g {\displaystyle {\mathfrak {g}}} is a complex semisimple Lie algebra with Cartan subalgebra h {\displaystyle {\mathfrak {h}}} ...

T—in terms of the highest weight of the representation. In the closely related representation theory of semisimple Lie algebras, the Weyl character formula...

to a module in representation theory of semisimple Lie algebras that generalizes the character of a finite-dimensional representation and is analogous...

{\displaystyle A} . Thus the Gelfand representation is injective if and only if A {\displaystyle A} is (Jacobson) semisimple. The Banach space A = L 1 ( R )...