4 1. PRELIMINARIES

The second property above allows one to choose a polarization of Φ, i.e., a

decomposition Φ = Φ+ ∪ Φ−, where Φ− = −Φ+. Elements of Φ+ (resp. Φ−) are

called positive (resp. negative) roots. The set Π = {α1,...,αr} of positive simple

roots is a maximal collection of positive roots that can not be represented as a

positive linear combination of other roots. Every positive root is a nonnegative

linear combination of elements of Π with integer coeﬃcients. The sum of these

coeﬃcients is called the height of the root.

The Cartan matrix of g is defined by

A = (aij)i,j=1

r

= ( αj|αi

)i,j=1r

and is characterized by the following properties:

(i) A is an integral matrix with 2’s on the diagonal and non-positive off-diagonal

entries.

(ii) A is symmetrizable, i.e. there exists an invertible diagonal matrix D (e.g.

D = diag ((αi,αi))i=1)

r

such that DA is symmetric. In particular, A is sign-

symmetric, that is, aijaji ≥ 0 and aijaji = 0 implies that both aij and aji are

zero.

(iii) DA defined as above is positive definite. This implies, in particular, that

aijaji 3.

The information contained in the Cartan matrix can also be encoded in the

corresponding Dynkin diagram, defined as a graph on r vertices with ith and jth

vertices joined by an edge of multiplicity aijaji. If aijaji 1 then an arrow is

added to the edge. It points to j if (αi,αi) (αj,αj).

A simple Lie algebra is determined, up to an isomorphism, by its Cartan matrix

(or, equivalently, its Dynkin diagram). Namely, it has a presentation with the so-

called Chevalley generators {hi,e±i}i=1

r

and relations

(1.6) [hi,hj] = 0, [hi,e±j] = ±cjie±j, [ej,e−j] = δijhj,

and

ade±iaij 1−

e±j = 0.

The latter are called the Serre relations. In terms of positive simple roots and

Chevalley generators, the elements of the Cartan matrix can be rewritten as aij =

αj(hi), i, j ∈ [1,r].

The classification theorem for simple Lie algebras states that every Dynkin

diagram is of one of the types described in Fig. 1.1.

It is clear from relations (1.6) that for every i ∈ [1,r], one can construct an

embedding ρi of the Lie algebra sl2 into g by assigning

ρi

1 0

0 1

= hi, ρi

0 1

0 0

= ei, ρi

0 0

1 0

= e−i,

which can be integrated to an embedding (also denoted by ρi) of the group SL2

into the group G. In particular,

(1.7) ρi

1 t

0 1

= exp(tei), ρi

1 0

t 1

= exp(te−i)

for any t ∈ C. One-parameter subgroups

(1.8) xi

±(t)

= exp(te±i)

generated by e±i will play an important role in what follows.