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the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups (and some...

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...

a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional...

mathematics, a Lie algebra (pronounced /liː/ LEE) is a vector space g {\displaystyle {\mathfrak {g}}} together with an operation called the Lie bracket, an...

exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which...

square root that is meant. The square roots of an integer are algebraic integers—more specifically quadratic integers. The square root of a positive integer...

from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation...

some positive integer n can be Lie groups (of course they must also have a differentiable structure). To every Lie group we can associate a Lie algebra whose...

used to read off the list of simple Lie algebras and Riemannian symmetric spaces. Together with the commutative Lie group of the real numbers, R {\displaystyle...

CSA, is a nilpotent subalgebra h {\displaystyle {\mathfrak {h}}} of a Lie algebra g {\displaystyle {\mathfrak {g}}} that is self-normalising (if [ X ,...

root α determines not just a 1-dimensional subspace of the Lie algebra of G, but also a copy of the additive group Ga in G with the given Lie algebra...

In algebra, the rational root theorem (or rational root test, rational zero theorem, rational zero test or p/q theorem) states a constraint on rational...

the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically...

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...

mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form...

The fundamental theorem of algebra, also called d'Alembert's theorem or the d'Alembert–Gauss theorem, states that every non-constant single-variable polynomial...

objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the...

In the theory of Lie groups, the exponential map is a map from the Lie algebra g {\displaystyle {\mathfrak {g}}} of a Lie group G {\displaystyle G} to...

mathematical theories of Lie groups and Lie algebras. For the topics in the representation theory of Lie groups and Lie algebras, see Glossary of representation...

Lie groups, Lie algebras and their representation theory, a Lie algebra extension e is an enlargement of a given Lie algebra g by another Lie algebra...