Maximal torus

In the mathematical theory of compact 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, abelian Lie subgroup of G (and therefore isomorphic to the standard torus Tn). A maximal torus is one which is maximal among such subgroups. That is, T is a maximal torus if for any torus T′ containing T we have T = T′. Every torus is contained in a maximal torus simply by dimensional considerations. A noncompact Lie group need not have any nontrivial tori (e.g. Rn). The dimension of a maximal torus in G is called the rank of G. The rank is well-defined since all maximal tori turn out to be conjugate. For semisimple groups the rank is equal to the number of nodes in the associated Dynkin diagram. The unitary group U(n) has as a maximal torus the subgroup of all diagonal matrices. That is, T is clearly isomorphic to the product of n circles, so the unitary group U(n) has rank n. A maximal torus in the special unitary group SU(n) ⊂ U(n) is just the intersection of T and SU(n) which is a torus of dimension n − 1. A maximal torus in the special orthogonal group SO(2n) is given by the set of all simultaneous rotations in any fixed choice of n pairwise orthogonal planes (i.e., two dimensional vector spaces). Concretely, one maximal torus consists of all block-diagonal matrices with diagonal blocks, where each diagonal block is a rotation matrix. This is also a maximal torus in the group SO(2n+1) where the action fixes the remaining direction. Thus both SO(2n) and SO(2n+1) have rank n. For example, in the rotation group SO(3) the maximal tori are given by rotations about a fixed axis. The symplectic group Sp(n) has rank n. A maximal torus is given by the set of all diagonal matrices whose entries all lie in a fixed complex subalgebra of H. Let G be a compact, connected Lie group and let be the Lie algebra of G. The first main result is the torus theorem, which may be formulated as follows: Torus theorem: If T is one fixed maximal torus in G, then every element of G is conjugate to an element of T.

About this result

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related publications (4)

Related concepts (13)

Related courses (3)

Related lectures (7)

Compact group

In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural generalization of finite groups with the discrete topology and have properties that carry over in significant fashion. Compact groups have a well-understood theory, in relation to group actions and representation theory. In the following we will assume all groups are Hausdorff spaces.

Weyl group

In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to the roots, and as such is a finite reflection group. In fact it turns out that most finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these.

Maximal torus

Embeddings of maximal tori in classical groups and explicit Brauer-Manin obstruction

Eva Bayer Fluckiger, Ting-Yu Lee

2018

Examples of algebraic groups of type G (2) having the same maximal tori

Ting-Yu Lee

Answering a question of A. Rapinchuk, we construct examples of non-isomorphic semisimple algebraic groups H (1) and H (2) of type G (2) having coherently equivalent systems of maximal k-tori.

Maik Nauka/Interperiodica/Springer2016

Embeddings of maximal tori in classical groups over fields of characteristic not 2 are the subject matter of several recent papers. The aim of the present paper is to give necessary and sufficient con

Turpion Ltd2016

On introduit les algèbres de Lie semisimples de dimension finie sur les nombres complexes et démontre le théorème de classification de celles-ci.

This course is an introduction to the theory of Riemann surfaces. Riemann surfaces naturally appear is mathematics in many different ways: as a result of analytic continuation, as quotients of complex

This course presents geometric constructions of irreducible representations of semi-simple Lie Algebras and their Weyl groups by means of Springer theory.

Equidistribution of CM Points

Explores the joint equidistribution of CM points for cubic fields, emphasizing periodic orbits and quadratic algebraic forms.

Bounding the Poisson bracket invariant on surfaces

Covers the concept of bounding the Poisson bracket invariant on surfaces, exploring joint work with A. Logunov and S. Tanny.

Local Homeomorphisms and Coverings MATH-410: Riemann surfaces

Covers the concepts of local homeomorphisms and coverings in manifolds, emphasizing the conditions under which a map is considered a local homeomorphism or a covering.