By Pierre-emmanuel Caprace
This paintings is dedicated to the isomorphism challenge for cut up Kac-Moody teams over arbitrary fields. This challenge seems to be a distinct case of a extra common challenge, which is composed in opting for homomorphisms of isotropic semi uncomplicated algebraic teams to Kac-Moody teams, whose photograph is bounded. given that Kac-Moody teams own typical activities on dual constructions, and because their bounded subgroups may be characterised by means of mounted aspect houses for those activities, the latter is absolutely a stress challenge for algebraic team activities on dual structures. the writer establishes a few partial tension effects, which we use to end up an isomorphism theorem for Kac-Moody teams over arbitrary fields of cardinality at the least four. specifically, he obtains a close description of automorphisms of Kac-Moody teams. this gives a whole knowing of the constitution of the automorphism staff of Kac-Moody teams over floor fields of attribute zero. an analogous arguments enable to regard unitary types of complicated Kac-Moody teams. particularly, the writer exhibits that the Hausdorff topology that those teams hold is an invariant of the summary crew constitution. ultimately, the writer proves the non-existence of co principal homomorphisms of Kac-Moody teams of indefinite variety over limitless fields with finite-dimensional goal. this gives a partial option to the linearity challenge for Kac-Moody teams
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Extra resources for Abstract homomorphisms of split Kac-Moody groups
For (v), notice ﬁrst that by deﬁnition, GH centralizes H. Let g ∈ CG (H). Then g acts on B H and on B¯H . By (ii), GH acts transitively on pairs of opposite chambers of B¯H . Hence there exists g ∈ GH such that g g ﬁxes a pair of opposite chambers of B¯H . This implies that g g ﬁxes a pair of opposite chambers of BH (see [Cap05b]). 5, we may assume that g g ﬁxes the standard twin apartment A. Since FixG (A) = T , we obtain g g ∈ T and hence g ∈ (g )−1 T ⊂ GH . 6(ii) can be interpreted as a combinatorial version of the fact that the centralizer of a diagonalizable subgroup of a reductive algebraic group is itself reductive.
U−i ) be the subring [n] [n] Zei (resp. n∈Z≥0 Zfi ) of u for λ ∈ Λ∨ and n ∈ Z≥0 . UgD and let U0 be the subring generated by all n Finally, let UD be the subring of UgD generated by U0 and all Ui and U−i for i ∈ I. 4]). 2. Deﬁnition of the adjoint representation. We keep the notation of the previous subsection. e. a commutative ring with a unit) and a subring A of UD , we set (A)R := A ⊗Z R. 1]), and the corresponding functor is abusively denoted by Aut(UD ). 20 3. KAC-MOODY GROUPS AND ALGEBRAIC GROUPS Let now F = (G, (ϕi )i∈I , η) be the basis of a Tits functor G of type D.
Thus (ii) is a straightforward consequence of (i). Recall that a diagonal automorphism of the group SL2 (K) is an automorphism of the form x → dxd−1 where d is a diagonal matrix of GL2 (K). 10. Let F and K be ﬁelds and suppose that |F| ≥ 4. If F is ﬁnite, suppose also char(F) = char(K). Let πF : SL2 (F) → ΓF and πK : SL2 (K) → ΓK be nontrivial surjective homomorphisms. Given a nontrivial group homomorphism ϕ : ΓF → ΓK there exists a ﬁeld homomorphism ζ : F → K, an inner automorphism ι and a diagonal automorphism δ of SL2 (K) such that the diagram: SL2 (ζ) SL2 (F) −−−−−−−−→ SL2 (K) ⏐ ⏐ ⏐ ⏐π ◦δ◦ι π F ΓF K ϕ −−−−−→ ΓK 24 3.
Abstract homomorphisms of split Kac-Moody groups by Pierre-emmanuel Caprace