Maximal Torus - Properties

Properties

Let G be a compact, connected Lie group and let 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 (cf. Cartan subalgebra)
  • Given a maximal torus T in G, every element g ∈ G 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 n − r is even.

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