CHAPTER 1

The objects: Kac-Moody groups, root data and

Tits buildings

1.1. Kac-Moody groups and Tits functors

1.1.1. Parameters of the construction. Let I be a finite set. A general-

ized Cartan matrix over I is a matrix A = (Aij )i,j∈I with integral coeﬃcients

such that

Aii = 2,

Aij ≤ 0 if i = j,

Aij = 0 ⇔ Aji = 0

for all i, j ∈ I. A (classical) Cartan matrix over I is a generalized Cartan matrix

over I which can be decomposed as a product of an invertible diagonal matrix and

a positive definite matrix. A generalized Cartan matrix is called symmetrizable

if it is the product of an invertible diagonal matrix and a symmetric matrix.

A Kac-Moody root datum is a system D = (I, A, Λ, (ci)i∈I , (hi)i∈I ) where

I is a finite set, A is a generalized Cartan matrix over I, Λ is a free Z-module, ci

is an element of Λ for each i ∈ I, hi is an element of the Z-dual Λ∨ of Λ and the

relation

ci|hj = Aji

holds for all i, j ∈ I. The Kac-Moody root datum D is called simply connected

if the hi’s form a basis of Λ∨.

1.1.2. Kac-Moody algebras. Let A = (Aij )i,j∈I be a generalized Cartan

matrix. The Kac-Moody algebra of type A over C is the complex Lie alge-

bra, noted gA, generated by the elements ei, fi and hi (i ∈ I) with the following

presentation (i, j ∈ I):

[hi, ej ] = Aij ej ,

[hi, fj ] = −Aijfj,

[hi, hj ] = 0,

[ei, fi] = −hi,

[ei, fj ] = 0 for i = j,

(ad

ei)−Aij +1(ej

) = ad

(fi)−Aij +1(fj

) = 0.

The Lie algebra gA is the derived algebra of the “classical” Kac-Moody algebra

g(A) considered by Kac in [Kac90]. This follows from Gabber-Kac’ theorem (see

[Kac90, Theorem 9.11]).

1.1.3. Existence and uniqueness of Tits functors. Throughout, we make

the convention that a ring is a commutative Z-algebra with a unit.

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