Abstract homomorphisms of split Kac-Moody groups by Pierre-emmanuel Caprace

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 different case of a extra normal challenge, which is composed in opting for homomorphisms of isotropic semi easy algebraic teams to Kac-Moody teams, whose photograph is bounded. seeing that Kac-Moody teams own traditional activities on dual structures, and because their bounded subgroups might be characterised via mounted element homes for those activities, the latter is admittedly a pressure challenge for algebraic staff 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. particularly, he obtains an in depth description of automorphisms of Kac-Moody teams. this offers an entire realizing of the constitution of the automorphism team of Kac-Moody teams over floor fields of attribute zero. an analogous arguments let to regard unitary types of advanced Kac-Moody teams. particularly, the writer indicates that the Hausdorff topology that those teams hold is an invariant of the summary staff constitution. eventually, the writer proves the non-existence of co imperative homomorphisms of Kac-Moody teams of indefinite sort over countless fields with finite-dimensional objective. this gives a partial strategy to the linearity challenge for Kac-Moody teams

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Let H be a commutative subgroup of G(K) and suppose that each element of H is contained in a K-split torus. Then G contains a maximal K-split torus T normalized by H. In particular T(K) contains a finite index subgroup of H. Moreover, if H is finite of order prime to the order of the Weyl group of G, then H is contained in T(K). Proof. 19). The version stated above can be obtained as follows. We work by induction on the dimension n of G. For n = 1, the identity component of G is a maximal K-split torus and the result is clear.

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. KAC-MOODY GROUPS AND ALGEBRAIC GROUPS commutes. Proof. The hypotheses imply that K has cardinality ≥ 4. F F Let U+ (resp.

Let ϕ : G → G be an isomorphism. Suppose that G is infinite and |K| ≥ 4. Then there exist an inner automorphism α of G , a bijection π : I → I and, for each i ∈ I, a field isomorphism ζi : K → K , a diagonal automorphism δi of SL2 (K ) and an automorphism ιi of SL2 (K ) which is either trivial or transposeinverse, such that the diagram SL2 (ζi ) SL2 (K) −−−−−→ SL2 (K ) ⏐ ⏐ ⏐ ⏐ϕ ϕi π(i) ◦δi ◦ιi α◦ϕ G −−−−→ G commutes for every i ∈ I. In particular, the bijection π extends to an isomorphism, also noted π, of the Weyl groups of G and G such that (ϕ, π) is an isomorphism of the twin root data associated with G and G .

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