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...

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...

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...

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

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...

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...

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

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

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;...

In mathematics, the special linear Lie algebra of order n {\displaystyle n} over a field F {\displaystyle F} , denoted s l n F {\displaystyle {\mathfrak...

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...

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...

system Φ {\displaystyle \Phi } , there exists a finite-dimensional semisimple Lie algebra whose root system is the given Φ {\displaystyle \Phi } . The theorem...

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

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...

simple Lie algebras. A semisimple algebraic group is a linear algebraic group whose radical of the identity component is trivial. Semisimple algebra Semisimple...

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...

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...

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...

simple Lie algebras Classical Lie algebras: Exceptional Lie algebras: semisimple 1.  A semisimple Lie group 2.  A semisimple Lie algebra is a nonzero Lie algebra...

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

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

of a Lie algebra or Lie group is an element whose centralizer has dimension as small as possible. For example, in a complex semisimple Lie algebra, an...

real semisimple Lie algebra g with Iwasawa decomposition g = k ⊕ a ⊕ n can be written as the sum of three commuting elements of the Lie algebra X = S...

weight classifies the irreducible representations of a complex semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} . There is a closely related theorem...