2 1. THE OBJECTS: KAC-MOODY GROUPS, ROOT DATA AND TITS BUILDINGS
Let D = (I, A, Λ, (ci)i∈I , (hi)i∈I ) be a Kac-Moody root datum. Let is the
split torus scheme, i.e. is the group functor on the category of rings defined
by TΛ(R) = Homgr(Λ,
R×).
With the datum D, J. Tits [Tit87b, §3.6] associates a system F =
(G, (ϕi)i∈I , η) consisting of:
a group functor G on the category of rings,
a collection (ϕi)i∈I of morphisms of functors ϕi : SL2 G,
a morphism of functors η : G,
which satisfies the following conditions, where
rhi
denotes the element λ r
λ,hi
of TΛ:
(KMG1): if K is a field, G(K) is generated by the images of ϕi(K) and
η(K);
(KMG2): for every ring R, the homomorphism η(R) : TΛ(R) G(R) is
injective;
(KMG3): for i I and r R×, one has ϕi
r 0
0
r−1
= η(rhi );
(KMG4): if ι is an injective homomorphism of a ring R in a field K, then
G(ι) : G(R) G(K) is injective;
(KMG5): there is a homomorphism Ad : G(C) Aut(gA) whose kernel is
contained in η(TΛ(C)), such that, for c C,
Ad ϕi
1 c
0 1
= exp ad cei,
Ad ϕi
1 0
c 1
= exp ad (−cfi),
and, for t TΛ(C),
Ad(η(t))(ei) = t(ci) · ei, Ad(η(t))(fi) = t(−ci) · fi.
The group functor G as above is called a Tits functor of type D and of
basis F. By definition, a (split) Kac-Moody group of type D over a field K is
the value on K of a Tits functor of type D. The main result of [Tit87b] asserts
that the restriction of G to the category of fields is completely characterized by the
conditions (KMG1)–(KMG5) modulo some additional non-degeneracy condition on
the images of the ϕi’s (see [Tit87b, Theorem 1] for a precise statement).
1.1.4. An alternative construction in the 2-spherical case. Let A be
a generalized Cartan matrix over a (finite) set I. For each subset J I, we set
AJ := (Aij )i,j∈J . The matrix A is called 2-spherical if for every 2-subset J of I
the matrix AJ is a (classical) Cartan matrix. Equivalently, A is 2-spherical if and
only if Aij Aji 3 for all i = j I.
In this section we present an explicit construction of Kac-Moody
groups of type D, where D = (I, A, Λ, (ci)i∈I , (hi)i∈I ) is the simply connected
Kac-Moody root datum associated with a 2-spherical generalized Cartan matrix A.
Let K be a field and assume that K is of cardinality at least 3 (resp. at least
4) if Aij = −2 (resp. Aij = −3) for some i, j I. For each i I, let Xi be a copy
of SL2(K) and for each 2-subset J = {i, j} of I, let Xi,j be a copy of the universal
Chevalley group of type AJ over K. Let also ϕi,j : Xi Xi,j be the canonical
monomorphism corresponding to the inclusion of Cartan matrices A{i} A{i,j}.
The direct limit of the inductive system formed by the groups Xi and Xi,j along
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