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...

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...

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)...

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...

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...

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...

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...

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

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...

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...

{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...

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...

algebraic torus (which is not necessarily compact, in contrast to a complex torus). A k-torus is a torus defined over k. The centralizer of a maximal torus is...

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...

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

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

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...

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...

over an algebraically closed field K {\displaystyle K} with a split maximal torus T {\displaystyle T} then its root datum is a quadruple ( X ∗ , Φ , X...

{\displaystyle W} is a Weyl group of G {\displaystyle G} corresponding to a maximal torus of B {\displaystyle B} . The Bruhat decomposition of G {\displaystyle...

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...

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...

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...

maximal invariant convex cone. Any intermediate cone is uniquely determined by its intersection with the Lie algebra of a maximal torus in a maximal compact...

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

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...

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...

(n)=\mathrm {Sp} (2g,\mathbb {R} )\cap \mathrm {GL} (g,\mathbb {C} )} is the maximal torus. Since the isometry group of a symmetric space G / K {\displaystyle...

Galois group on a Dynkin diagram. All split groups (those with a split maximal torus) are quasi-split. These correspond to quasi-split groups where the action...

characteristic 2 type, the width is usually the rank (the dimension of a maximal torus of the algebraic group). The quasithin groups were classified in a 1221-page...