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.
1
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