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...

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...

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

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...

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...

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...

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...

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...

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...

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

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}...

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

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

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...

non-identity proper subgroup is a maximal nilpotent subgroup, but only the subgroup of order three is a Fischer subgroup (Wehrfritz 1999, p. 98). Cartan subalgebra...

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...

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...

the general linear group GL(4, R), of which the Lorentz group is a Cartan subgroup. The geometric equivalence principle postulating the existence of a...

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...

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

mathematical fields of Lie theory and algebraic topology, the notion of Cartan pair is a technical condition on the relationship between a reductive Lie...

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

mathematical physics, nonlinear realization of a Lie group G possessing a Cartan subgroup H is a particular induced representation of G. In fact, it is a representation...

has an element that generates a dense subgroup. (In particular, a torus is topologically cyclic.) Cartan subgroup Procesi 2007, Ch. 4. § 2.. Knapp 2001...

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...