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 non-zero...

representation is semisimple. Every reductive Lie algebra is isomorphic to the product of an abelian Lie algebra and a semisimple Lie algebra. For example...

of representation theory, a Lie algebra representation or representation of a Lie algebra is a way of writing a Lie algebra as a set of matrices (or endomorphisms...

every semisimple Lie group is a central product of simple Lie groups. The semisimple Lie groups are exactly the Lie groups whose Lie algebras are semisimple...

algebras is one of the major achievements of Wilhelm Killing and Élie Cartan. A direct sum of simple Lie algebras is called a semisimple Lie algebra....

algebras. The best known example is the monster Lie algebra. Finite-dimensional semisimple Lie algebras have the following properties: They have a nondegenerate...

semi-simple Lie algebra g {\displaystyle {\mathfrak {g}}} over a field of characteristic 0 {\displaystyle 0} . In a finite-dimensional semisimple Lie algebra over...

language of algebraic geometry. Just as complex semisimple Lie algebras are classified by Dynkin diagrams, the real forms of a semisimple Lie algebra are classified...

their algebraic properties (abelian; simple; semisimple). For more examples of Lie groups and other related topics see the list of simple Lie groups;...

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...

the semisimple Lie algebras form two large and generally complementary classes, as is shown by the Levi decomposition. The solvable Lie algebras are precisely...

affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much...

the Lie algebra is the Lie algebra of a compact semisimple Lie group. In general, the Lie algebra of a compact Lie group decomposes as the Lie algebra direct...

split real form of a complex Lie algebra, and because split semisimple Lie algebras (more generally, split reductive Lie algebras) over any field share many...

coefficients of the characteristic equation of a regular semisimple element of a Lie algebra are invariant under the adjoint group, from which it follows...

Hopf algebras depending on an auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or...

This (real) Lie algebra has dimension n2 − 1. More information about the structure of this Lie algebra can be found below in § Lie algebra structure. In...

Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify...

representation theory of semisimple Lie algebras is one of the crowning achievements of the theory of Lie groups and Lie algebras. The theory was worked...

graded algebra under the bracket operation. A choice of Cartan decomposition endows any semisimple Lie algebra with the structure of a graded Lie algebra. Any...

These algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to the structure of a Lie algebra such...

mathematics, a semisimple algebra is an associative Artinian algebra over a field which has trivial Jacobson radical (only the zero element of the algebra is in...

of the Lie algebra of K is semisimple. Conversely, every complex semisimple Lie algebra has a compact real form isomorphic to the Lie algebra of a compact...

examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the...

of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups...

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

derivation.) Let g {\displaystyle {\mathfrak {g}}} be a complex semisimple Lie algebra and h {\displaystyle {\mathfrak {h}}} a Cartan subalgebra of g {\displaystyle...

representation is well understood is that of semisimple (or reductive) Lie groups, where the associated Lie algebra representation forms a (g,K)-module. Examples...

theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a...

(Lie group) Simple Lie group Compact Lie group, Compact real form Semisimple Lie algebra Root system Simply laced group ADE classification Maximal torus...