Jacques Tits

The Book of Involutions

This monograph yields a comprehensive exposition of the theory of central simple algebras with involution, in relation with linear algebraic groups. It aims to provide the algebra-theoretic foundations for much of the recent work on linear algebraic groups over arbitrary fields. Involutions are viewed as twisted forms of similarity classes of hermitian or bilinear forms, leading to new developments on the model of the algebraic theory of quadratic forms. Besides classical groups, phenomena related to triality are also discussed, as well as groups of type F_4 or G_2 arising from exceptional Jordan or composition algebras. Several results and notions appear here for the first time, notably the discriminant algebra of an algebra with unitary involution and the algebra-theoretic counterpart to linear groups of type D_4. For research mathematicians and graduate students working in central simple algebras, algebraic groups, nonabelian Galois cohomology or Jordan algebras.

Free subgroups in linear groups

(i) H contains no non-abelian free group. (ii) G has a solvable normal subgroup R such that G,‘R is locally .fnite (i.e., every jinite subset generates a

Groupes réductifs sur un corps local : I. Données radicielles valuées

nite subgroup). (iii) G possesses a subgroup G’ of finite index such that if V’ denotes any composition factor of the k[G’]-module V and k’ the endomorphism ring of V’ (i.e., the centralizer of G’ in End,L V’), then k’ is a jield and V’ has a k’-basis with respect to which the matrices representing the elements of G’ are scalar multiples (by elements of k’) f o matrices whose entries are algebraic over the prime field of k.

A Local Approach to Buildings

© Publications mathématiques de l’I.H.É.S., 1972, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Sur le groupe des automorphismes d’un arbre

The object of this paper is the comparison of two notions of (combinatorial) buildings, that of [14] (or [1]), and an earlier version (cf. e.g. [10]), which has lately regained interest through the work of F. Buekenhout on sporadic groups (cf. [2], [3], [16]).

On Generalized Hexagons and a Near Octagon whose Lines have Three Points

Le but de cet article est l’etude de la structure du groupe des automorphismes d’un arbre, c’est-a-dire d’un graphe connexe sans circuit. Les principaux resultats obtenus sont resumes dans l’enonce suivant.

A “theorem of Lie-Kolchin” for trees

Proofs are given of the facts that any finite generalized hexagon of order (2, t) is isomorphic to the classical generalized hexagon associated with the group G2(2) or to its dual if t = 2 and that it is isomorphic to the classical generalized hexagon associated with the group 3D4(2) if t = 8. Furthermore, it is shown that any near octagon of order (2, 4; 0, 3) is isomorphic to the known one associated with the sporadic simple group HJ.

Sur les constantes de structure et le théorème d'existence des algèbres de Lie semi-simples

Publisher Summary This chapter reviews a theorem of Lie-Kolchin for trees. It is known that the automorphism group of a tree T behaves, in some respects, as SL 2 over a field with a non-archimedean valuation. In that analogy, the end space of T plays the role of the projective line. The Lie-Kolchin theorem, as generalized a la Rosenlicht, asserts that a connected k -split solvable subgroup of SL n ( k ), for any field k , has a fixed point in the projective space P n-1 ( k ). Thus, a solvable fixed point free automorphism group of a tree T leaves invariant, an end or a pair of ends of T, can be regarded as an analogue of the theorem of Lie-Kolchin for trees.. The main application of the above result aims at showing that the group G of rational points of an algebraic, almost simple, group of relative rank ≥2 over a field k cannot operate without fixed point or fixed end on a tree. The study of groups that cannot operate on trees without fixed point was initiated by J.-P. Serre.. For trees in the usual sense, the theorem of Lie-Kolchin under review is implicitly contained in a paper of H. Bass.

Strongly inner anisotropic forms of simple algebraic groups

© Publications mathématiques de l’I.H.É.S., 1966, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Groupes algébriques simples sur un corps local

(the definition of the index of a semisimple algebraic group over a field will be recalled in 1.55). More precisely, we show, in Proposition 2(B) (cf. 3.1), that if, over a given field k, there exists a central division algebra of degree 4 (i.e., of dimension 16) and of order 4 in the Brauer group, then there exists a group of type Ei;T over a purely transcendental extension of degree 1 k(t) of k. This will be deduced in two different ways from Theorem 1 of Section 2, whose precise statement is too technical to be given in this introduction, but whose spirit is that it enables one, from the existence of groups of a given type with preassigned properties (of a geometric and cohomological nature) over a field k, to infer the existence of groups of a more complicated type, meeting similar requirements, over k(t). That theorem has other, interesting applications to classification problems. For instance, we also deduce from it, again in two different ways (Proposition 3(B) in 3.2), that the existence of a division algebra of degree 3 over k implies the existence of groups with index