nLab
maximal torus

Contents

Context

Group Theory

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • 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 }

          </semantics></math></div>

          Models

          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)

          \infty-Lie theory

          ∞-Lie theory (higher geometry)

          Background

          Smooth structure

          Higher groupoids

          Lie theory

          ∞-Lie groupoids

          ∞-Lie algebroids

          Formal Lie groupoids

          Cohomology

          Homotopy

          Examples

          \infty-Lie groupoids

          \infty-Lie groups

          \infty-Lie algebroids

          \infty-Lie algebras

          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 , prop. 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 .

          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.