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On 2-spherical Kac-Moody groups and their central extensions

Pierre-Emmanuel Caprace

2007
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Forum mathematicum
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We provide a new presentation for simply connected Kac-Moody groups of 2spherical type and for their universal central extensions. Under mild local restrictions, these results extend to the more general class of groups of Kac-Moody type (i.e. groups endowed with a root datum). Let D = (G, (U α ) α∈Φ ) be a twin root datum of type (W, S), let Π be a basis of the root system Φ, let H := α∈Φ N G (U α ) and, for α, β ∈ Π, let X α := U α ∪ U −α , X α,β := X α ∪ X β and L α := HX α , L α,β := HX α,β
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... We will need to consider the following condition, which excludes the possibility for X α,β to be isomorphic to some very small finite groups: The Curtis-Tits theorem asserts that, if the Coxeter group W is finite, then G is the amalgamated sum of its subgroups L α and L α,β , where α, β run over Π. By the main result of [AM97], this remains true when the Coxeter system (W, S) is 2-spherical (but W possibly infinite), provided that D satisfies Condition (Co * ). The following theorem, which will be proved in Section 3.3.2 below, is a refinement of the latter fact. Theorem A. Suppose that (W, S) is 2-spherical and D satisfies Condition (Co * ). Let G be the direct limit of the inductive system formed by the X α and X α,β with natural inclusions (α, β ∈ Π). Then the kernel of the canonical homomorphism G → G is central. In the special case where W is finite, this is the main result of [Tim04]. However, our approach, which builds upon P. Abramenko's ideas outlined in §2 of [AM97] (see Theorem 3.6 below), is quite different from the one in [Tim04] and provides a considerably shorter proof of that result. We also note that the main result of [AM97] recalled above can be easily deduced from Theorem A (see Corollary 3.8 below). The kernel of the canonical homomorphism G → G might be infinite in general. Actually, in the case where G is a split Kac-Moody group (in the sense of [Ti87]), the group G is always a quotient of the simply connected central extension of G with finite kernel (see Proposition 3.12 below). Furthermore, if G itself is a simply connected split Kac-Moody group, then G and G coincide (see Corollary 3.13). As an application of this observation, we now describe an elementary and handy way to define a simply connected split Kac-Moody group of 2-spherical type starting from its Dynkin diagram. Application. Let I = {1, 2, . . . , n}, let A = (A ij ) i,j∈I be a generalized Cartan matrix. This means that A ii = 2, that A ij ∈ Z ≤0 and that A ij = 0 if and only if A ji = 0 for all i = j ∈ I. For each subset J ⊂ I, we set A J := (A ij ) i,j∈J . Assume that the generalized Cartan matrix is 2-spherical. In other words, for each 2-subset J of I we require that A J is a classical Cartan matrix or, equivalently, that A ij A ji ≤ 3 for all i = j ∈ I. Note that the information contained in such a generalized Cartan matrix is equivalent to the information contained in the associated Dynkin diagram. Let K be a field and assume that K is of cardinality at least 3 (resp. at least 4) if A ij = −2 (resp. A ij = −3) for some i, j ∈ I. For each i ∈ I, let X i be a copy of SL 2 (K) and for each 2-subset J = {i, j} of I, let X i,j be a copy of the universal Chevalley group of type A J over K. Let also ϕ i,j : X i → X i,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 X i and X i,j along with the monomorphisms ϕ i,j (i, j ∈ I) coincides with the simply connected Kac-Moody group of type A over K. J (see §4 in [St68]). Let now K := Ker ϕ ∩ G sc J | J ⊂ I, |J| = 1 or 2 . By definition (since I is finite), the group K is a finite central subgroup of G sc . Furthermore, we have clearly K ∩ G sc J = Ker ϕ ∩ G sc J = Ker ϕ ∩ G sc J for every J ⊂ I such that |J| = 1 or 2. Therefore, we have K = Ker ϕ.

doi:10.1515/forum.2007.031
fatcat:vq7hn45p25cjhc6g7ew26vvirq