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 settling on homomorphisms of isotropic semi easy algebraic teams to Kac-Moody teams, whose snapshot is bounded. due to the fact that Kac-Moody teams own traditional activities on dual constructions, and because their bounded subgroups might be characterised by way of mounted aspect houses for those activities, the latter is admittedly 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 no less than four. specifically, he obtains an in depth description of automorphisms of Kac-Moody teams. this gives a whole figuring out of the constitution of the automorphism team of Kac-Moody teams over flooring fields of attribute zero. an identical arguments permit to regard unitary kinds of complicated Kac-Moody teams. particularly, the writer exhibits that the Hausdorff topology that those teams hold is an invariant of the summary staff constitution. ultimately, the writer proves the non-existence of co valuable homomorphisms of Kac-Moody teams of indefinite variety over endless fields with finite-dimensional goal. this offers a partial strategy to the linearity challenge for Kac-Moody teams

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**Abstract homomorphisms of split Kac-Moody groups - download pdf or read online**

This paintings is dedicated to the isomorphism challenge for break up Kac-Moody teams over arbitrary fields. This challenge seems to be a unique case of a extra normal challenge, which is composed in settling on homomorphisms of isotropic semi easy algebraic teams to Kac-Moody teams, whose snapshot is bounded. given that Kac-Moody teams own typical activities on dual structures, and because their bounded subgroups will be characterised by means of fastened aspect homes for those activities, the latter is admittedly a stress challenge for algebraic team activities on dual constructions.

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Let Z = (G, (Uα )α∈Φ ) be a twin root datum and B be the associated twin building. A diagonalizable subgroup of G is called regular (with respect to Z) if BH is reduced to a single twin apartment. Thus, roughly speaking a diagonalizable subgroup is regular if its ﬁxed point set is as small as possible. We will come back to this notion a little further. At this point, we just record the following. 6. Let Z = (G, (Uα )α∈Φ ) be the twin root datum associated to a Kac-Moody group G over the ﬁeld K.

Proof. 3] for (i), (ii) and (iii). For Assertion (iv), note that the ‘only if’ part is clear. Let now s¯ := AdK (s)|W ¯ In view of and suppose s¯ semisimple. 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.

Suppose that J is spherical. Then, for P J = LJ ∈ {+, −}, we have UJ. Denoting by RJ the unique J-residue of B stabilized by P J , we have J J LJ = P+J ∩ P−J = StabG (R+ ) ∩ StabG (R− ) and the group U J acts regularly on the J-residues opposite RJ in B. Proof. 2]. The group U J is called the unipotent radical of P J and LJ is called a Levi factor (or a Levi subgroup). 2. Levi decomposition of bounded subgroups. In this section we describe a Levi decomposition for bounded subgroups. 3] under a technical assumption, called (NILP).