1.2. SIMPLE LIE ALGEBRAS AND GROUPS 3

1.2. Simple Lie algebras and groups

1.2.1. Let G be a connected simply-connected complex Lie group with an iden-

tity element e and let g

∼

= TeG be the corresponding Lie algebra. We will often use

notation G = exp g and g = Lie G. The dual space to g will be denoted by

g∗

and

the value of ∈

g∗

at ξ ∈ g will be denoted by , ξ .

Recall the definitions of the adjoint (resp. co-adjoint) actions of G and g on g

(resp. g∗):

Adx ξ =

d

dt

(

x−1

exp(ξt)x

)

t=0

, adη ξ = [η, ξ]

and

Adx

∗

, ξ = , Adx ξ , adη

∗

, ξ = , adη ξ

for any ξ ∈ g.

The Killing form on g is a bilinear symmetric form (·, ·)g defined by

(1.4) (ξ, η)g = tr(adξ adη),

where adξ adη is viewed as a linear operator acting in the vector space g.

1.2.2. Let now g be a complex simple Lie algebra of rank r with a Cartan

subalgebra h = a ⊕ ia. Recall that h is a maximal commutative subalgebra of g

of dimension r, and that the adjoint action of h on g can be diagonalized. More

precisely, for α ∈

h∗,

define

gα = {ξ ∈ g : [h, ξ] = α(h)ξ for any h ∈ h}.

Clearly, g0 = h. A nonzero α such that gα = 0 is called a root of g, and a collection

Φ of all roots is called the root system of g. Then:

(i) For any α ∈ Φ, dim gα = 1, and

(ii) g has a direct sum decomposition

g = h ⊕ (⊕α∈Φgα) ,

which is graded by Φ, that is, [gα, gβ] = gα+β, where the right hand side is zero if

α + β is not a root.

The Killing form (1.4) is nondegenerate on the simple algebra g, and so is its

restriction to h, which we will denote by (·, ·). Thus h∗ and h can be identified via

(·, ·) :

h∗

α → hα ∈ h : α(h) = (hα,h)

for all h ∈ h. Moreover, the restriction of (·, ·) to the real vector space spanned

by hα,α ∈ Φ is real-valued and positive definite. Thus, the real vector space

h0∗

spanned by α ∈ Φ is a real Euclidean vector space with the inner product (·, ·).

In particular, a reflection with respect to the hyperplane normal to α ∈ h0

∗

can be

defined by

(1.5) sαβ = β − β|α α,

where β|α = 2

(β,α)

(α,α)

.

We can now summarize the properties of the root system Φ:

(i) Φ contains finitely many nonzero vectors that span

h0;∗

(ii) for any α ∈ Φ, the only multiples of α ∈ Φ are ±α;

(iii) Φ is invariant under reflections sα, α ∈ Φ;

(iv) for any α, β ∈ Φ, the number β|α is an integer.