Maximal torus

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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 other 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.

Examples[edit]

The unitary group U(n) has as a maximal torus the subgroup of all diagonal matrices. That is,

T = \left\{\mathrm{diag}(e^{i\theta_1},e^{i\theta_2},\dots,e^{i\theta_n}) : \forall j, \theta_j \in \mathbb R\right\}.

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 n pairwise orthogonal 2-planes. 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.

Properties[edit]

Let G be a compact, connected Lie group and let \mathfrak g be the Lie algebra of G.

  • A maximal torus in G is a maximal abelian subgroup, but the converse need not hold.
  • The maximal tori in G are exactly the Lie subgroups corresponding to the maximal abelian, diagonally acting subalgebras of \mathfrak g (cf. Cartan subalgebra)
  • Given a maximal torus T in G, every element gG is conjugate to an element in T.
  • Since the conjugate of a maximal torus is a maximal torus, every element of G lies in some maximal torus.
  • All maximal tori in G are conjugate. Therefore, the maximal tori form a single conjugacy class among the subgroups of G.
  • It follows that the dimensions of all maximal tori are the same. This dimension is the rank of G.
  • If G has dimension n and rank r then nr is even.

Weyl group[edit]

Given a torus T (not necessarily maximal), the Weyl group of G with respect to T can be defined as the normalizer of T modulo the centralizer of T. That is, W(T,G) := N_G(T)/C_G(T). Fix a maximal torus T = T_0 in G; then the corresponding Weyl group is called the Weyl group of G (it depends up to isomorphism on the choice of T). The representation theory of G is essentially determined by T and W.

  • The Weyl group acts by (outer) automorphisms on T (and its Lie algebra).
  • The centralizer of T in G is equal to T, so the Weyl group is equal to N(T)/T.
  • The identity component of the normalizer of T is also equal to T. The Weyl group is therefore equal to the component group of N(T).
  • The normalizer of T is closed, so the Weyl group is finite
  • Two elements in T are conjugate if and only if they are conjugate by an element of W. That is, the conjugacy classes of G intersect T in a Weyl orbit.
  • The space of conjugacy classes in G is homeomorphic to the orbit space T/W and, if f is a continuous function on G invariant under conjugation, the Weyl integration formula holds:
\displaystyle{\int_G f(g)\, dg = |W|^{-1} \int_T f(t) |\Delta(t)|^2\, dt,}
where Δ is given by the Weyl denominator formula.

See also[edit]

References[edit]

  • Adams, J. F. (1969), Lectures on Lie Groups, University of Chicago Press, ISBN 0226005305 
  • Bourbaki, N. (1982), Groupes et Algèbres de Lie (Chapitre 9), Éléments de Mathématique, Masson, ISBN 354034392X 
  • Dieudonné, J. (1977), Compact Lie groups and semisimple Lie groups, Chapter XXI, Treatise on analysis 5, Academic Press, ISBN 012215505X 
  • Duistermaat, J.J.; Kolk, A. (2000), Lie groups, Universitext, Springer, ISBN 3540152938 
  • Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN 0821828487 
  • Hochschild, G. (1965), The structure of Lie groups, Holden-Day