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The building of a semi-simple algebraic group

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Part of the Lecture Notes in Mathematics book series (LNM,volume 386)

Abstract

The following notations and terminology will be used in § 5 and, for some of them, throughout the sequel of the notes: if X is an algebraic group defined over some field k, the group of k -rational points of X over k is denoted by X(k) (instead of Xk, as in [8]; the notations Z( ), R( ), Ru( ) stand for “centralizer of”, “radical of”, “unipotent radical of” respectively; the group of all automorphisms (resp. all special automorphisms) of a building Δ is represented by Aut Δ (resp. Spe Δ); in these notes, the qualificatives “reductive” and “semi-simple” are meant to imply “connected”; if G is a reductive group, T a maximal torus, Φ = Φ(T,G) the corresponding root system, Ψ ⊂ Φ a basis of Φ (“system of simple roots”), we call Dynkin diagram of G the diagram over Ψ (in the sense of 2.11) defined in the usual way together with an orientation of each multiple stroke, telling which extremity of this stroke has the smallest length; when we want to disregard these orientations, we talk about the diagram underlying the Dynkin diagram.

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© 1974 Springer-Verlag Berlin Heidelberg

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(1974). The building of a semi-simple algebraic group. In: Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Mathematics, vol 386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-38349-9_5

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  • DOI: https://doi.org/10.1007/978-3-540-38349-9_5

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-06757-3

  • Online ISBN: 978-3-540-38349-9

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