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
= g : [h, ξ] = α(h)ξ for any h h}.
Clearly, g0 = h. A nonzero α such that = 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 = 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 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.
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