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...
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In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by G m {\displaystyle \mathbf {G} _{\mathbf {m} }} , G m {\displaystyle...
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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...
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Compact group (section Maximal tori and root systems)
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...
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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)...
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algebraically closed) field k {\displaystyle k} is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent...
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{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...
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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...
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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...
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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...
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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...
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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...
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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...
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{\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...
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torus Umbilic torus Genus-2 surface, also called "double torus" Maximal torus Clifford torus Torus palatinus, a bony growth on the palate Torus mandibularis...
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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...
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a maximal torus in the general linear group (and are their own centralizer), the generalized permutation matrices are the normalizer of this torus, and...
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Toral subalgebra (redirect from Maximal toral subalgebra)
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...
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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...
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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...
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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...
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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...
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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...
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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...
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Semisimple Lie algebra Root system Simply laced group ADE classification Maximal torus Weyl group Dynkin diagram Weyl character formula Representation of a...
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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...
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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...
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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...
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Orthogonal group (section Maximal tori and Weyl groups)
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...
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homogeneous space. If G is a compact, connected Lie group, it contains a maximal torus T and the space G/T of left cosets with the quotient topology is a compact...
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