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 basic challenge, which is composed in settling on homomorphisms of isotropic semi basic algebraic teams to Kac-Moody teams, whose picture is bounded. due to the fact that Kac-Moody teams own average activities on dual constructions, and because their bounded subgroups could be characterised by means of fastened element homes for those activities, the latter is basically a tension challenge for algebraic crew activities on dual structures. the writer establishes a few partial tension effects, which we use to turn out an isomorphism theorem for Kac-Moody teams over arbitrary fields of cardinality not less than four. particularly, he obtains a close description of automorphisms of Kac-Moody teams. this gives a whole knowing of the constitution of the automorphism team 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 indicates 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 imperative homomorphisms of Kac-Moody teams of indefinite variety over countless fields with finite-dimensional aim. this offers a partial method to the linearity challenge for Kac-Moody teams
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Additional resources for Abstract homomorphisms of split Kac-Moody groups
Then s¯ is contained in a maximal torus of L. 7(ii) and the functoriality of the adjoint representation, ¯ which is Ad ¯ -diagonalizable and one sees that there exists an element s ∈ G(K) K such that AdK¯ (s )|WK¯ = s¯. It follows from (iii) that (s )−1 s is AdK¯ -diagonalizable ¯ and is and centralizes s . Therefore s is an AdK¯ -diagonalizable element of G(K) thus AdK -semisimple. 4. Link between the adjoint representation and the action on the twin building. 5, the kernel of the adjoint action of G(K) and its action on the associated twin building coincide.
It suﬃces to prove the following claim: If two chambers x, y are ﬁxed by H, then there exists a twin apartment all of whose chambers are ﬁxed by H. By hypothesis, H ﬁxes some twin apartment A of B. Let f (x) (resp. f (y)) be the numerical distance between x (resp. y) and A. The proof of the claim above is by induction on f (x) + f (y). The result is clear for f (x) + f (y) = 0. Now assume that f (x) + f (y) > 0. Without loss of generality we have f (y) > 0. Considering a gallery of minimal possible length joining y to a chamber of A, we obtain a chamber y adjacent to y such that f (x) + f (y ) < f (x) + f (y).
17 below and rests heavily on Tits’ rigidity theorem for actions on trees. However, this proposition requires in turn to get a sharp control on diagonalizable subgroups of Kac-Moody groups, which is a rather delicate problem. A whole section of this chapter is devoted to diagonalizable subgroups; another one is concerned with a slightly larger class of subgroups called completely reducible. The chapter ends with the main technical auxiliary to the proof of the isomorphism theorem. 1. 1. The statement.
Abstract homomorphisms of split Kac-Moody groups by Pierre-emmanuel Caprace