Maximal torus information

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)...

{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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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

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...

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...

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...

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