# nLab maximal torus

Contents

group theory

### Cohomology and Extensions

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

#### $\infty$-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

$\infty$-Lie groupoids

$\infty$-Lie groups

$\infty$-Lie algebroids

$\infty$-Lie algebras

# Contents

## Definition

Let $G$ be a compact Lie group.

A torus in $G$ is an abelian subgroup $T \hookrightarrow G$ which is connected (and compact). Note that any compact connected abelian Lie group must be a torus, hence the name.

A maximal torus is a subgroup which is maximal with these properties.

The Lie algebra of a maximal torus of $G$ is a Cartan subalgebra of the Lie algebra of $G$.

## Properties

Suppose throughout that the compact Lie group $G$ is connected.

###### Proposition

Given a choice of maximal torus $T \hookrightarrow G$, then each element $g \in G$ is conjugate to an element of its maximal torus, i.e. there exists $h_g \in G$ such that $h_g g h_g^{-1} \in T \hookrightarrow G$.

(e.g. Johansen, theorem 2.7.3)

###### Remark

For the unitary group $U(n)$, by example , prop. is the spectral theorem that unitary matrices may be diagonalized.

###### Corollary

Any two choices of maximal tori are conjugate.

###### Proposition

Any two elements of a maximal torus $T\hookrightarrow G$ are conjugate in $G$ precisely if they are conjugate via the Weyl group.

(e.g. Johansen, prop 2.7.13)

###### Corollary

The conjugacy classes of $G$ are in bijection to $T/W(G,T)$.

###### Proposition

Under the assumptions on $G$, the exponential map $\exp: \mathfrak{g} \to G$ is surjective.

###### Proof

The exponential map $\mathbb{R} \to S^1$ is surjective; similarly, if $\mathfrak{h}$ is a Cartan subalgebra of $\mathfrak{g}$, then the exponential map $\mathfrak{g} \to G$ maps $\mathfrak{h}$ homomorphically onto a maximal torus. Also the exponential map $\mathfrak{g} \to G$ preserves conjugation by any element of $G$, i.e., $\exp(g x g^{-1}) = g\exp(x)g^{-1}$. Then surjectivity follows from Proposition .

## Examples

###### Example

The maximal torus of a unitary group $U(n)$ is the subgroup $U(1)^n\hookrightarrow U(n)$ of diagonal matrices with unitary entries.

###### Example

The maximal torus of the special unitary group $SU(n+1)$ is again $U(1)^n$ (one dimension lower) included as the subgroup of diagonal matrices whose first $n$ diagonal entries are arbitrary elements of $U(1)$ and whose last diagonal entry is the inverse of their product.

###### Example

The maximal torus of the special orthogonal group $SO(2n)$ and $SO(2n+1)$ is $U(1)^n \simeq SO(2)^n$ included block-diagonal (with, in the odd-dimensional case, the remaining block entry being the unit).

e.g.

• Troels Roussauc Johansen, section 2.6 of Character Theory for Finite Groups and Compact Lie Groups pdf