CHAPTER 1

Lie-Theoretic Preliminaries

The purpose of this chapter is to recall some basic Lie theoretic results and to

establish some associated terminology and notation.

Weyl chambers

When one replaces the torus Tn in the action-angle model space Tn x Rn with

a compact connected Lie group G, the natural generalization of the action space

M.n

turns out to be a Weyl chamber of G. We now recall the standard 'geometric'

definition of this object.

1.1 Definition. If a group G acts on a manifold X, then an orbit in X is regular

if there exist no orbits in X of strictly greater dimension. A point x G l i s called

regular if it lies on a regular orbit. Let Q denote the Lie algebra of a compact

connected Lie group G and let t C Q be any maximal Abelian subalgebra. Denote

by 9reg C Q the regular points of the adjoint action of G, g • £ = Ad^ £ (£ € fl). A

connected component to of the set t Pi 0reg is called an (open) Weyl chamber of G

in fl.

Some related Lie theoretic facts needed in the sequel are summarized below.

1.2 Theorem. Let G be a compact connected Lie group. Then:

1- Breg C £ is open and dense.

2. All maximal tori of G are conjugate.

3. Every g G G lies in some maximal torus.

4. The Lie algebra t of any maximal torus T C G is a maximal Abelian subal-

gebra.

5. Every point £ G Q belongs to at least one maximal Abelian subalgebra.

6. The map sending a maximal torus to its Lie algebra is a bijection between

the maximal tori of G and the maximal Abelian subalgebras of Q. If t is a

maximal Abelian subalgebra, then t D greg C t is open and dense, and the

isotropy group G^ is identical for all points £ G i fl greg. This group G$

is precisely the inverse oft under the above correspondence, i.e., G^ is the

unique maximal torus whose Lie algebra is t.

7. Each regular adjoint orbit intersects each Weyl chamber in exactly one point.

8. If i is a maximal Abelian subalgebra and we define

fr1

= [g, t], then one has

the direct sum decomposition

g = t e t

±

.

Proofs of the above facts can be found in, e.g., Brocker and torn Dieck (1985).

The reader may view these facts as generalizations to compact groups of familiar

properties of the rotation group SO(3). For example, 1.2.3 corresponds to Euler's

theorem that every rotation is a rotation about some fixed axis.

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