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maximal torus

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id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

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Contents

Definition

Let GG be a compact Lie group.

A torus in GG is an abelian subgroup TGT \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 GG is called a Cartan subalgebra of the Lie algebra of GG.

Properties

Suppose throughout that the compact Lie group GG is connected.

Proposition

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

(e.g. Johansen, theorem 2.7.3)

Remark

For the unitary group U(n)U(n), by example 1, prop. 1 is the spectral theorem that unitary matrices may be diagonalized.

Corollary

Any two choices of maximal tori are conjugate.

(e.g. Johansen, corollary 2.7.5)

Proposition

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

(e.g. Johansen, prop 2.7.13)

Corollary

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

Proposition

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

Proof

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

Examples

Example

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

Example

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

Example

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

References

e.g.

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

Last revised on September 6, 2016 at 18:00:43. See the history of this page for a list of all contributions to it.