0.2. Root Systems 3

example, a Chevalley basis has very simple structure constants, all in Z,

though it is not usually essential to make this kind of refined choice; see [46,

VIII, 2.4] (where the convention is that [xaya] = —ha), [125, §25].

0.2. Root Systems

The occurrence of root systems in the Lie algebra setting has led to a some-

what more widely applicable notion of abstract root system $ in a vector

space over K. Ultimately the classification turns out to be the usual one

in terms of Dynkin diagrams, but the axiomatic treatment suggests useful

generalizations and is logically independent of the Lie algebra theory.

In the structure theory of g, the root system spans a Q-form E$ of the

dual space f)*, where the Killing form is nondegenerate. Thus E := R®q Eo

has a natural structure of euclidean space. Humphreys [125, §9] then defines

an abstract root system in a finite dimensional euclidean space E over R,

with inner product denoted (A,//), to be a finite set $ of vectors spanning

E and not containing 0. It is required that for each a G fr, Ra fl $ = {±a}.

Moreover, the euclidean reflection sa defined by A i- A — 2(A, a)/(a, a)a

sends $ to itself. Further, the Cartan invariant (/?, a

v

) := 2(/3, a)/(a, a)

lies in Z for all a,/3 G J . Here a

v

:= 2a/(a, a) is the coroot of a. The

Z-span Ar of $ in E is called the root lattice. (The notation in [125] is a

little different.)

It is this formulation which we adopt, identifying EQ with the Q-span of

the roots in f)*, which also contains the root lattice Ar.

In Bourbaki [45, VI, §1] (where the notation and initial viewpoint are

different), one starts just with a finite subset f r as above in a finite dimen-

sional vector space E over R. It is required that each a G $ determine a

unique coroot

av

in E*, so that the "reflection" sa?av permutes $. Here

5a,av(A)

= A — (A,

av)a.

Further, the values of each a

v

on $ must lie in Z.

In either version one gets a dual root system

frv

:= {a

v

| a G E } (living

in the euclidean space E in our version, but in the dual space E* in the

Bourbaki version). This is a root system whose Dynkin diagram is dual to

that of $; in particular, if $ is of type B^, then $

v

is of type C^. In the

Killing form identification of f ) and f)*, the coroot a

v

corresponds to ha G f):

(/?,av)

=/3(ha) for a l l / 3 G * .

Each choice of simple system A in $ defines a partition into subsets

of positive and negative roots (denoted respectively $

+

and $~). Here A

forms a basis of E (or a Z-basis of Ar), while each (3 G $

+

can be written

uniquely as /3 = EaeA

c

«

a w

^ h c

a

G Z

+

. Define the height of (3 to be

ht j3 :=

X]Q:€A

C

OL\

so ht /? = 1 if and only if ft G A.