In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group G {\displaystyle G} over a (not necessarily algebraically closed)...

In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra h {\displaystyle {\mathfrak {h}}} of a Lie algebra g {\displaystyle...

form a subgroup called the torsion subgroup. Cartan subgroup Fitting subgroup Fixed-point subgroup Fully normalized subgroup Stable subgroup Gallian...

subalgebra is called a parabolic Lie algebra. Hyperbolic group Cartan subgroup Mirabolic subgroup A. Borel (2001). Essays in the History of Lie Groups and Algebraic...

closed-subgroup theorem (sometimes referred to as Cartan's theorem) is a theorem in the theory of Lie groups. It states that if H is a closed subgroup of...

Cartan's theorem may refer to several mathematical results by Élie Cartan: Closed-subgroup theorem, 1930, that any closed subgroup of a Lie group is a...

In mathematics, the Cartan decomposition is a decomposition of a semisimple Lie group or Lie algebra, which plays an important role in their structure...

In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information...

as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Cartan connection. Cartan reformulated...

rank. We call B the (standard) Borel subgroup, T the (standard) Cartan subgroup, and W the Weyl group. A subgroup of G is called parabolic if it contains...

cardinality of a generating set for the group Rank of a Lie group – see Cartan subgroup Rank of a matroid, the maximal size of an independent set Rank of a...

the intended meaning (and in fact maximal proper subgroups are not in general compact). The Cartan-Iwasawa-Malcev theorem asserts that every connected...

map Cartan matrix Cartan pair Cartan subalgebra Cartan subgroup Cartan's method of moving frames Cartan's theorem, a name for the closed-subgroup theorem...

subgroup is SU(2) × SU(2)/(−1,−1). It has a non-algebraic double cover that is simply connected. The Dynkin diagram for G2 is given by . Its Cartan matrix...

In abstract algebra, the Cartan–Brauer–Hua theorem (named after Richard Brauer, Élie Cartan, and Hua Luogeng) is a theorem pertaining to division rings...

Any topologically closed subgroup of a Lie group is a Lie group. This is known as the closed subgroup theorem or Cartan's theorem. The quotient of a...

e^{i\theta }} and e − i θ {\displaystyle e^{-i\theta }} . Compact group Cartan subgroup Cartan subalgebra Toral Lie algebra Bruhat decomposition Weyl character...

this work was later perfected by Élie Cartan. The final classification is often referred to as Killing-Cartan classification. Unfortunately, there is...

corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into...

as a compact homogeneous space K/T, where T = K ∩ B is a (compact) Cartan subgroup of K. An integral weight λ determines a G-equivariant holomorphic line...

named by analogy with the theory of tori in Lie group theory (see Cartan subgroup). For example, over the complex numbers C {\displaystyle \mathbb {C}...

Generalized flag variety, Bruhat decomposition, BN pair, Weyl group, Cartan subgroup, group of adjoint type, parabolic induction Real form (Lie theory)...

algebra of SL(2, R) so that iH generates the Lie algebra of a compact Cartan subgroup K (so in particular unitary representations split as a sum of eigenspaces...

integration against an analytic function on the regular set. If H is a Cartan subgroup of G and H' is the set of regular elements in H, then Θ π | H ′ = ∑...

torus defined over k. The centralizer of a maximal torus is called a Cartan subgroup. Diagonal subgroup Borel, A. Linear algebraic groups, 2nd ed. v t e...

by Cartan, and two exceptional cases; the classification can be deduced from Borel–de Siebenthal theory, which classifies closed connected subgroups containing...

In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up...

In mathematics, the Cartan–Ambrose–Hicks theorem is a theorem of Riemannian geometry, according to which the Riemannian metric is locally determined by...

operation is matrix multiplication. The special unitary group is a normal subgroup of the unitary group U(n), consisting of all n×n unitary matrices. As a...

its maximal compact subgroup K acts by conjugation on the component P in the Cartan decomposition. If A is a maximal Abelian subgroup of G contained in...