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)...

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 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...

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

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

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...

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

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...

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

finite-dimensional representations of semisimple Lie algebras Fulton, William; Harris, Joe (1991). Representation theory. A first course. Graduate Texts...

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...

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

of the representation theory of Lie groups, the study of representations of SU(2) is fundamental to the study of representations of semisimple Lie groups...

element is semisimple if its image under the adjoint representation is semisimple; see Semisimple Lie algebra#Jordan decomposition. This disambiguation...

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

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

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...

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

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

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...

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

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

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

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

theory of semisimple Lie algebras (or the closely related representation theory of compact Lie groups), the weights of the dual representation are the negatives...

Lorentz group is deduced using the general framework of the representation theory of semisimple Lie algebras. The finite-dimensional representations of the...