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 common challenge, which is composed in picking out homomorphisms of isotropic semi basic algebraic teams to Kac-Moody teams, whose photo is bounded. in view that Kac-Moody teams own ordinary activities on dual structures, and because their bounded subgroups will be characterised by way of fastened aspect houses for those activities, the latter is really a tension challenge for algebraic crew activities on dual constructions. the writer establishes a few partial pressure effects, which we use to end up an isomorphism theorem for Kac-Moody teams over arbitrary fields of cardinality no 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 identical arguments let to regard unitary types of advanced Kac-Moody teams. specifically, the writer indicates that the Hausdorff topology that those teams hold is an invariant of the summary workforce constitution. ultimately, the writer proves the non-existence of co crucial homomorphisms of Kac-Moody teams of indefinite variety over endless fields with finite-dimensional objective. this offers a partial technique to the linearity challenge for Kac-Moody teams

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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 common challenge, which is composed in picking out homomorphisms of isotropic semi basic algebraic teams to Kac-Moody teams, whose photo is bounded. in view that Kac-Moody teams own ordinary activities on dual structures, and because their bounded subgroups will be characterised by way of fastened aspect houses for those activities, the latter is really a tension challenge for algebraic crew activities on dual constructions. the writer establishes a few partial pressure effects, which we use to end up an isomorphism theorem for Kac-Moody teams over arbitrary fields of cardinality no 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 identical arguments let to regard unitary types of advanced Kac-Moody teams. specifically, the writer indicates that the Hausdorff topology that those teams hold is an invariant of the summary workforce constitution. ultimately, the writer proves the non-existence of co crucial homomorphisms of Kac-Moody teams of indefinite variety over endless fields with finite-dimensional objective. this offers a partial technique to the linearity challenge for Kac-Moody teams

Show description

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Proof. 2]. 3. Bounded subgroups as algebraic groups. Let D be a Kac-Moody root datum, G be a Tits functor of type D and K be a field. A subgroup H of G := G(K) is called AdK -locally finite (resp. AdK -locally unipotent) if every vector of the K-vector space (UD )K is contained in a finitedimensional subspace V invariant under AdK (H) (resp. and such that AdK (H)|V is a unipotent subgroup of GL(V )). A subgroup H of G is called AdK -diagonalizable (resp. AdK -semisimple) if (UD )K decomposes into a direct sum of one-dimensional subspaces invariant under AdK (H) (resp.

Similar phenomena occur for all types of Kac-Moody groups. We do not want to go into details on this topic; relevant related results may be found in [H´ ee90] and [Cho00]. In the present section, we merely illustrate by an example that over finite fields, the abstract structure of Kac-Moody groups might contain only very poor information on their defining generalized Cartan matrices. n Given integers m, n ∈ Z≥0 , we denote by Dm the simply connected Kac-Moody 2 −m over root datum associated with the generalized Cartan matrix −n 2 n n n I = {1, 2}.

5, the kernel of the adjoint action of G(K) and its action on the associated twin building coincide. Actually, the relationship between these actions is very sharp. The present subsection aims to bring this relationship into focus. 7. Let K be an infinite field, G be a Tits functor and H be a subgroup of G := G(K). We have the following: (i) H is bounded if and only if it is AdK -locally finite. 2) if and only if it is AdK diagonalizable. (iii) If H is bounded unipotent then it is AdK -locally unipotent.

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