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...

twice through the circle, the surface is a spindle torus (or self-crossing torus or self-intersecting torus). If the axis of revolution passes through the...

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

R2, the Clifford torus is an embedded torus in R2 × R2 = R4. If R4 is given by coordinates (x1, y1, x2, y2), then the Clifford torus is given by x 1 2...

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...

Look up torus in Wiktionary, the free dictionary. A torus, pl. tori, is a type of surface. Torus may also refer to: Torus (album), a 2013 album by Sub...

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...

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...

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

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...

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...

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 =...

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...

{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...

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...

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...

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...

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...

variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends...

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...

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...

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

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

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...

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...

In mathematics, a complex torus is a particular kind of complex manifold M whose underlying smooth manifold is a torus in the usual sense (i.e. the cartesian...

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...

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...