Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group G is a compact, connected...

In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by G m {\displaystyle \mathbf {G} _{\mathbf {m} }} , G m {\displaystyle...

is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis...

torus theorem which states that every element of K {\displaystyle K} belongs to a maximal torus and that all maximal tori are conjugate. The maximal torus...

given a torus T < G (which need not be maximal), the Weyl group with respect to that torus is defined as the quotient of the normalizer of the torus N = N(T)...

characters of an F-stable maximal torus, which is irreducible (up to sign) when the character is in general position. When the maximal torus is split, these representations...

For a linear algebraic group G, a maximal torus in G means a torus in G that is not contained in any bigger torus. For example, the group of diagonal...

algebraically closed) field k {\displaystyle k} is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent...

{h}}} may be taken as the complexification of the Lie algebra of a maximal torus of the compact group. If g {\displaystyle {\mathfrak {g}}} is a linear...

with a, b, c, d real and ad − bc = 1. A maximal compact connected abelian Lie subgroup, or maximal torus T, is given by the subset of all matrices of...

is a Borel subgroup of G W is a Weyl group of G corresponding to a maximal torus of B. The Bruhat decomposition of G is the decomposition G = B W B =...

Here the group B is a Borel subgroup and N is the normalizer of a maximal torus contained in B. The notion was introduced by Armand Borel, who played...

contains a split maximal torus T over k (that is, a split torus in G whose base change to k ¯ {\displaystyle {\overline {k}}} is a maximal torus in G k ¯ {\displaystyle...

group with maximal torus T. Hopf showed that the centralizer of a torus S ⊆ T is a connected closed subgroup containing T, so of maximal rank. Indeed...

torus Umbilic torus Genus-2 surface, also called "double torus" Maximal torus Clifford torus Torus palatinus, a bony growth on the palate Torus mandibularis...

Stated another way, a torus embedded in R3 is an asymmetric reduced-dimension projection of the maximally symmetric Clifford torus embedded in R4. The relationship...

that K contains a maximal torus of G, so has maximal rank. On the other hand, the centralizer of the subgroup generated by the torus S of elements exp...

that K contains a maximal torus of H, so has maximal rank. On the other hand, the centralizer of the subgroup generated by the torus S of elements exp...

semisimple element, say x; the linear span of x is then a toral subalgebra. Maximal torus, in the theory of Lie groups Humphreys 1972, Ch. II, § 8.1. Proof (from...

group E8 has dimension 248. Its rank, which is the dimension of its maximal torus, is eight. Therefore, the vectors of the root system are in eight-dimensional...

representations if and only if it has the same rank as a maximal compact subgroup K. In other words, a maximal torus T in K must be a Cartan subgroup in G. (This...

a maximal torus in the general linear group (and are their own centralizer), the generalized permutation matrices are the normalizer of this torus, and...

classes) were reflections of the classical linear groups and their maximal torus structure. What is more, the Chern class itself was not so new, having...

orthogonal circles that do not twist around each other, and so form a maximal torus within the Lie group, corresponding to a collection of R mutually-commuting...

up to an action of the maximal torus: the flag corresponds to the Borel group, and the inner product corresponds to the maximal compact subgroup. The stabilizer...

is the standard one-dimensional torus. In O(2n) and SO(2n), for every maximal torus, there is a basis on which the torus consists of the block-diagonal...

Semisimple Lie algebra Root system Simply laced group ADE classification Maximal torus Weyl group Dynkin diagram Weyl character formula Representation of a...

parabolic subgroup of G, and N = N(K), where N is the normalizer of a split maximal torus contained in P. In particular, any finite group of Lie type has the...

uses the machinery of the maximal torus and the associated root system. Specifically, let T {\displaystyle T} be a maximal torus in a connected compact Lie...

determining the branching rule when the groups G and H share a common maximal torus: in this case the Weyl group of H is a subgroup of that of G, so that...