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Topological groups, Lie groups Definition

Maximal torus

Builds on Compact Lie group Torus as a group

Definition

Let $G$ be a compact Lie group. A torus in $G$ is a compact connected abelian Lie subgroup of $G$ . Each such subgroup is isomorphic to the torus group $\mathbb{T}^k$ for some $k \ge 0$ . A maximal torus of $G$ is a torus $T \subseteq G$ such that every torus $T'$ with $T \subseteq T' \subseteq G$ satisfies $T' = T$ .

Remarks

Example: in $G = U(n)$ , the subgroup $T$ of diagonal matrices $\operatorname{diag}(z_1, \dots, z_n)$ with $\abs{z_j} = 1$ is a maximal torus. It is a torus, isomorphic to $\mathbb{T}^n$ . Let $T' \supseteq T$ be a torus. Each element of $T'$ commutes with a diagonal matrix with distinct entries, so it is diagonal; thus $T' = T$ . Non-example: in $U(2)$ , the subgroup $\set{\operatorname{diag}(z, 1) : \abs{z} = 1}$ is a torus, but it is not maximal, since it lies in the diagonal torus $\mathbb{T}^2$ . The dimension of a maximal torus is the rank of $G$ ; for $U(n)$ it is $n$ , and for $SU(n)$ it is $n - 1$ . Every torus of $G$ lies in some maximal torus, by a dimension argument.

Reading path

To understand this page from the ground up, read these 72 pages in order. The foundations come first.

Binary operation definition
Compact Lie group concept
Element definition
Group concept
Pontryagin dual concept
Proposition concept
Real numbers object
Commutativity definition
Set definition
Conjunction definition
Disjunction definition
Implication definition
Predicate definition
Empty set object
Family of sets definition
Ordered pair definition
Abelian group definition
Intersection definition
Union definition
Set-builder notation concept
Existential quantifier definition
Universal quantifier definition
Cartesian product definition
Circle group object
Subset definition
Union of a family of sets definition
Uniqueness quantifier definition
Binary relation definition
Universal set definition
Topological space definition
Reflexive relation definition
Symmetric relation definition
Transitive relation definition
Complement definition
Function definition
Basis of a topology definition
Subspace topology definition
Equivalence relation definition
Product topology definition
Preimage of a set definition
Metric space definition
Codomain definition
Function composition definition
Injective function definition
Equivalence class definition
Open ball definition
Surjective function definition
Metric topology definition
Quotient set definition
Bijective function definition
Open set definition
Quotient topology definition
Inverse function definition
Hausdorff space definition
Continuous map of topological spaces definition
Open cover definition
Closed set definition
Complement of an open set is closed statement
Homeomorphism definition
Compact space definition
Universal property of the quotient topology statement
Continuity via closed sets statement
Compact set definition
Compactness via open covers statement
Continuous image of a compact set is compact statement
Compact subsets of a Hausdorff space are closed statement
Closed subsets of compact sets are compact statement
Continuous bijection from a compact space to a Hausdorff space is a homeomorphism statement
Circle as a quotient of the real line example
Torus example
Torus as a group example
Maximal torus definition

Further reading

Web https://en.wikipedia.org/wiki/Maximal_torus

About this page

Tag	`Z2AJFZQTWT`
URI	`m2://Z2AJFZQTWT`
Kind	definition
Writing	W0 stub
MSC	22E15, 22C05
Defines	Maximal torus
Uses	Compact Lie group, Torus as a group

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