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Jacques Tits (1930–2021) was a Belgian-French mathematician renowned for his fundamental contributions to group theory and geometry. In particular, he introduced the concept of “buildings,” geometric structures that unify the study of groups. He also formulated the “Tits alternative,” which states that any finitely generated linear group is either virtually solvable or contains a free subgroup on two generators. Tits’s work profoundly influenced the classification of algebraic groups and the geometry of group actions. He received several prestigious honors, including the Wolf Prize in 1993 and, together with John G. Thompson, the Abel Prize in 2008. Tits spent much of his career at the Collège de France, and his ideas remain a cornerstone of modern algebraic and geometric research.

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Article Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque. was published on January 1, 1971 in the journal Journal für die reine und angewandte Mathematik (volume 1971, issue … Article Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque. was published on January 1, 1971 in the journal Journal für die reine und angewandte Mathematik (volume 1971, issue 247).
The combinatorial objects called buildings were first introduced (cf.) to provide a geometric approach to complex simple Lie groups – in particular the exceptional ones – and later on, more … The combinatorial objects called buildings were first introduced (cf.) to provide a geometric approach to complex simple Lie groups – in particular the exceptional ones – and later on, more generally, to isotropic algebraic simple groups (cf. eg.). The buildings which do arise in that way are spherical, that is, have finite Weyl groups. This assertion has a partial converse, proved in: roughly speaking, there is a one-to-one correspondence between the (isomorphism classes of) buildings of irreducible spherical type and rank r ≥ 3 and the algebraic absolutely simple groups of relative rank r, where the notion of algebraic simple groups must be suitably extended so as to include, for instance, classical groups over division rings of infinite dimension over their centre. In order to have a similar statement in the rank 2 case, one must impose an extra condition, the Moufang condition, on the buildings under consideration.
1 ) Le référée (outre plusieurs remarques dont nous le remercions) nous a signalé l'article de P. GÉRARDIN, Immeuble des groupes linéaires généraux, publié dans Non Commutative Harmonie Analysis and … 1 ) Le référée (outre plusieurs remarques dont nous le remercions) nous a signalé l'article de P. GÉRARDIN, Immeuble des groupes linéaires généraux, publié dans Non Commutative Harmonie Analysis and Lie Groups, Lecture Notes, 880, Springer 1981, qui nous avait échappé.On y trouve notamment (sous Fhypothcsc K localement compact) une étude des normes contenant nos résultats de 1.7 à 1.9 et une construction géométrique directe de la structure d'immeuble sur l'ensemble ^Ldes normes.
On donne dans le cas d'un groupe classique G des constructions concrètes.c'est-à-dire liées à la représentation naturelle de G, de l'immeuble et des schémas en groupes de la théorie générale … On donne dans le cas d'un groupe classique G des constructions concrètes.c'est-à-dire liées à la représentation naturelle de G, de l'immeuble et des schémas en groupes de la théorie générale des groupes réductifs sur un corps local ([3], [4]).Le cas des groupes linéaires généraux ou spéciaux ayant été traité dans une première partie [5].on s'intéresse aux autres groupes, c'est-à-dire aux groupes unitaires.ABSTRACT.-Thé building and group-schemes assodated by thé général theory ([3], [4]) to a reductive group over a local field are given a concrète interprétation in thé case of a classical group G; hère, "concrète" means "relatcd to thé natural représentation of G".Thé case of thé général and spécial linear groups bas been handied in a first part [5]; hère.we deal with thé other classical groups.i. e. thé unitary groups.
Let G be a finite simple group and let F be an algebraically closed field. A faithful projective F -representation of G of smallest possible degree often cannot be lifted … Let G be a finite simple group and let F be an algebraically closed field. A faithful projective F -representation of G of smallest possible degree often cannot be lifted to an ordinary representation of G, though it can of course be lifted to an ordinary representation of some central extension of G. It is a natural question to ask whether by considering non-central extensions, it is possible in some cases to decrease the smallest degree of a faithful projective representation.
Let u: G -> A be a differentiable representation of a Lie group into a 6-algebra.The differential u 0 = du e of u at the neutral element e of … Let u: G -> A be a differentiable representation of a Lie group into a 6-algebra.The differential u 0 = du e of u at the neutral element e of G is a representation of the Lie algebra 9 of G into A. Because a Lie group is locally the union of one-parameter subgroups and since the infinitesimal generator of a differentiable (multiplicative) sub-semi-group of A determines this sub-semi-group, the representation u 0 determines u if G is connected.We shall be concerned with the converse: given a representation u 0 of o, when can it be obtained by differentiating a representation u of G? We shall assume G connected and simply connected, which means that we are only interested in the local aspect of the problem.Call aeA integrable if a differentiable r: R-*A can be found such that r(s + t) = r(s)r(t) and r'(0) = a.We can only hope to integrate u 0 : g -» A to a differentiable u: G -> A if u o x is integrable for all XGQ.We shall prove the THEOREM.
Cette note a pour objet l’étude, faite d’un point de vue essentiellement géométrique, du plan projectif des octaves de Cayley et, plus particulièrement, du groupe des « collinéations » (automorphismes) … Cette note a pour objet l’étude, faite d’un point de vue essentiellement géométrique, du plan projectif des octaves de Cayley et, plus particulièrement, du groupe des « collinéations » (automorphismes) de ce plan, et du sous-groupe des collinéations qui laissent invariante une « conique d’Hermite ».
Introduction.Let A be the affine building of a simple adjoint algebraic group Q of relative rank > 2 over a locally compact local field K. Let Aut A (resp.E Aut … Introduction.Let A be the affine building of a simple adjoint algebraic group Q of relative rank > 2 over a locally compact local field K. Let Aut A (resp.E Aut A) denote the group of type-preserving (resp. of all) automorphisms of A. Note that E Aut A contains the group $(K) of ÜT-rational points of §.We will be interested in discrete subgroups of Aut A which are chamber-transitive on A. It is extremely rare that such groups exist and, as can therefore be expected, exceptions are interesting phenomena; our purpose is to list them all (see the theorem below).In order to describe them we must first introduce some notation.Let ƒ be a quadratic form in n variables over Q p with coefficients in Z.We let Pfi(/, Z[l/p]) denote the intersection PSO(/, Q p )'nPGL(n, Z[l/p]) within PGL(n,Q p ), and similarly PGO(/, Z[l/p]) = PGO(/,Q p ) nPGL(n,Z[l/p]).In the following list, T will always be a chamber-transitive subgroup of Aut A. The fundamental quadratic form (over Z) of the root lattice of type A n , B n , E n , normalized so that the long roots have squared length 2, will be denoted by a n ,6 n ,e n , respectively; note that b n is Yl" x l-(i) Let ƒ = eg, &7,ci6,66*^6* or as, and let A be the affine building of PSO(/,Q 2 ).Here T can be any group between r min = Pfi(/,Z[l/2]) and r ma x = PGO(/,Z[l/2]) fi Aut A. The quotient r max /r min is elementary abelian of order 1, 1, 1, 4, 2, or 2, respectively, and r max is generated by Train and reflections.(ii) Let ƒ = &5,e6, or b' e = Xa x ?+ ^xh an( l let A be the building of PSO(/,Q 3 ).The group r max (/) = PGO(/, Z[l/3]) n Aut A has 3, 5, or 9 conjugacy classes of chamber-transitive subgroups T. Passage mod 2 maps r m ax(b5) onto the symmetric group S5, and the preimages in r max (&5) of S5, A5, or a group of order 20 form the 3 desired conjugacy classes of groups T. The forms e& and b' e are rationally equivalent, and hence the buildings they define over Q3 are the "same" ; with suitable identifications of buildings and groups, T b = r max (ee) H r max (6é) has index 27 in r max (e6) and index 2 in r m ax(&6)-Passage mod 2 maps r max (e6) onto PGO(5,3), and the preimages in r max (e6) of the 5 different classes of flag-transitive subgroups of PGO(5,3) (cf.[S]) form the 5 desired conjugacy classes of groups I\ exactly 3 of which have members in T b .The 6 remaining conjugacy classes of chamber-transitive subgroups of r max (&6) not having members in r max (e6) consist of groups having index 1 or 2 in (r, r) for one of the chamber-transitive subgroups T of r b , where r is the reflection x$ »-• -XQ.
As a first step in the classification of all thick Moufang polygons, it is shown that every root ray datum of type A2, B2 or G2 has a filtration by … As a first step in the classification of all thick Moufang polygons, it is shown that every root ray datum of type A2, B2 or G2 has a filtration by an ordinary root datum.
[73] Originally published in C. R. Acad. Sci. Paris Ser. A Math. 263 (1966), 867–869. Reused with permission. [73] Originally published in C. R. Acad. Sci. Paris Ser. A Math. 263 (1966), 867–869. Reused with permission.
[80] Originally published in Algebraic geometry: papers presented at the Bombay Colloquium 1968, Tata Inst. Fund. Res. Stud. Math. 4, Oxford University Press, Oxford (1969), 75–82. Reused with permission. [80] Originally published in Algebraic geometry: papers presented at the Bombay Colloquium 1968, Tata Inst. Fund. Res. Stud. Math. 4, Oxford University Press, Oxford (1969), 75–82. Reused with permission.
L'auteur fait le point (en 1970) des connaissances relatives aux homomorphismes et automorphismes des groupes classiques. Il pose le probleme sous une forme tres generale: on a deux corps commutatifs … L'auteur fait le point (en 1970) des connaissances relatives aux homomorphismes et automorphismes des groupes classiques. Il pose le probleme sous une forme tres generale: on a deux corps commutatifs K,K′, un schema en groupes KG[K′G′] sur K[K′], un sous-groupe H[H′] du groupe des points de KG[K′G′] rationnels sur K[K′]. L'auteur decrit d'abord un type d'homomorphisme α:H→H′ qu'il appelle semi-algebrique. On le definit en considerant d'une part un homomorphisme de corps σ:K→K′; si KσG est le schema en groupes sur K′ deduit de KG par le changement de base σ, on a un homomorphisme canonique σ∗ du g certaines conditions, est tel que β(h)=α(h)χ(h), ou α est semi-algebrique et χ un homomorphisme de H dans le centre de H′. Il ne faut s'attendre a une reponse raisonnable que lorsque KG et K′G′ sont des schemas en groupes semi-simples, comme le montrent des exemples pathologiques donnes par l'auteur a la fin de son expose.Les exemples enumeres par l'auteur ou le probleme precedent a une reponse affirmative sont d'une part ceux examines par O'Meara et son ecole par la methode de caracterisation des rotations planes, developpee par O'Meara depuis 1966; d'autre part, ceux qui font l'object des travaux de A. Borel et de l'auteur. Ces derniers s'appliquent a tous les groupes algebriques absolument presque simples (classiques ou exceptionnels'') pourvuqu'ils soient isotropes''; par contre cette condition n'intervient pas dans la methode de O'Meara, mais cette derniere est limitee aux groupes classiques.
[13] Originally published in Bull. Soc. Math. Belg. 5 (1952), 44–52. Reused with permission. [13] Originally published in Bull. Soc. Math. Belg. 5 (1952), 44–52. Reused with permission.
Publications Tits J., Œuvres, Collected Works, 4 volumes, Buekenhout F., Mühlherr B., Tignol J.-P. et Maldeghem H. van (éd.), Zürich, European Mathematical Society, 2013. Tits J., Résumés des cours au … Publications Tits J., Œuvres, Collected Works, 4 volumes, Buekenhout F., Mühlherr B., Tignol J.-P. et Maldeghem H. van (éd.), Zürich, European Mathematical Society, 2013. Tits J., Résumés des cours au Collège de France 1973-2000, Doc. Math. 12, Paris, Société mathématique de France, 2013.
The combinatorial objects called buildings were first introduced (cf.) to provide a geometric approach to complex simple Lie groups – in particular the exceptional ones – and later on, more … The combinatorial objects called buildings were first introduced (cf.) to provide a geometric approach to complex simple Lie groups – in particular the exceptional ones – and later on, more generally, to isotropic algebraic simple groups (cf. eg.). The buildings which do arise in that way are spherical, that is, have finite Weyl groups. This assertion has a partial converse, proved in: roughly speaking, there is a one-to-one correspondence between the (isomorphism classes of) buildings of irreducible spherical type and rank r ≥ 3 and the algebraic absolutely simple groups of relative rank r, where the notion of algebraic simple groups must be suitably extended so as to include, for instance, classical groups over division rings of infinite dimension over their centre. In order to have a similar statement in the rank 2 case, one must impose an extra condition, the Moufang condition, on the buildings under consideration.
这篇结伴式文章多侧面展现了Armand Borel,他于2003年8月11日去世.8位作者记述他,依次是Serre,Chandrasekharan,Bombieri,Hirzebruch,Springer,Tits,Arthur,Prasad.Borel在代数和拓扑方面的研究足够好,使他在34岁时被聘为IAS(高等研究院)的数学教授.Serre综览他的数学全貌,而Hirzebruch侧重于拓扑方面以不同方式描述他的数学.Springer和Tits讨论Borel在代数群方面的工作,而Arthur评论算术群,以及Borel在这个领域的工作如何奠定自守形式现代理论的基石出.一位研究院的同事说道Borel强烈相信数学的统一性和书面记载的重要性.实现这些信念的方法包括各种角色:编辑、作者、教育家和会议组织者,本文作者中有几位详细描述了这些活动.Borel投入大量的努力到Bourbaki全集的撰稿中,Borel在1998年3月《通报》(Notices of the AMS)里文章“与Nicolas Bourbaki的25年,1949—1973”详细描写了他的经历.广泛认为Bourbaki关于Lie群和Lie代数这一章的写作Borel起了主要作用,这一章已经显出特别持久的价值.Borel是1962—79年间的Annals of Mathematics的编辑,1979—93年间Inventiones Mathematicae的编辑,以及其它杂志的编辑.1998-2000年间他悄悄地以无头衔的副主编方式为《通报》服务,在各种事务上向主编进言,尤其是纪念文章和《通报》与其它国家相应的杂志合作.Borel在策划关于A.Weil的各种文章以及纪念J.Leray和A.Lichnerowicz的文章,起了大而又默默无闻的作用.本文作者中几位描写到Borel著或编的一些书,这些书的列表中共有17本,除了他的文集:论文集【文集】,列在下面方框里.多数书是讨论班的结果,有时与他人合作有时没有.不管讨论班采取何种形式,人们都可相信Borel是每个人的导师.特别的注记是两次美国数学会暑期学校的会议录,1965年在Boulder[3]和1977年在Corvalli 这篇结伴式文章多侧面展现了Armand Borel,他于2003年8月11日去世.8位作者记述他,依次是Serre,Chandrasekharan,Bombieri,Hirzebruch,Springer,Tits,Arthur,Prasad.Borel在代数和拓扑方面的研究足够好,使他在34岁时被聘为IAS(高等研究院)的数学教授.Serre综览他的数学全貌,而Hirzebruch侧重于拓扑方面以不同方式描述他的数学.Springer和Tits讨论Borel在代数群方面的工作,而Arthur评论算术群,以及Borel在这个领域的工作如何奠定自守形式现代理论的基石出.一位研究院的同事说道Borel强烈相信数学的统一性和书面记载的重要性.实现这些信念的方法包括各种角色:编辑、作者、教育家和会议组织者,本文作者中有几位详细描述了这些活动.Borel投入大量的努力到Bourbaki全集的撰稿中,Borel在1998年3月《通报》(Notices of the AMS)里文章“与Nicolas Bourbaki的25年,1949—1973”详细描写了他的经历.广泛认为Bourbaki关于Lie群和Lie代数这一章的写作Borel起了主要作用,这一章已经显出特别持久的价值.Borel是1962—79年间的Annals of Mathematics的编辑,1979—93年间Inventiones Mathematicae的编辑,以及其它杂志的编辑.1998-2000年间他悄悄地以无头衔的副主编方式为《通报》服务,在各种事务上向主编进言,尤其是纪念文章和《通报》与其它国家相应的杂志合作.Borel在策划关于A.Weil的各种文章以及纪念J.Leray和A.Lichnerowicz的文章,起了大而又默默无闻的作用.本文作者中几位描写到Borel著或编的一些书,这些书的列表中共有17本,除了他的文集:论文集【文集】,列在下面方框里.多数书是讨论班的结果,有时与他人合作有时没有.不管讨论班采取何种形式,人们都可相信Borel是每个人的导师.特别的注记是两次美国数学会暑期学校的会议录,1965年在Boulder[3]和1977年在Corvalli
Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes … Les Elements de mathematique de Nicolas Bourbaki ont pour objet une presentation rigoureuse, systematique et sans prerequis des mathematiques depuis leurs fondements. Ce premier volume du Livre sur les Groupes et algebre de Lie, neuvieme Livre du traite, est consacre aux concepts fondamentaux pour les algebres de Lie. Il comprend les paragraphes: - 1 Definition des algebres de Lie; 2 Algebre enveloppante d une algebre de Lie; 3 Representations; 4 Algebres de Lie nilpotentes; 5 Algebres de Lie resolubles; 6 Algebres de Lie semi-simples; 7 Le theoreme d Ado. Ce volume est une reimpression de l edition de 1971.
By A. Borel: pp. x, 398; Cloth $12.50, Paper $4.95. (W. A. Benjamin, Inc., New York, 1969). By A. Borel: pp. x, 398; Cloth $12.50, Paper $4.95. (W. A. Benjamin, Inc., New York, 1969).
Article Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque. was published on January 1, 1971 in the journal Journal für die reine und angewandte Mathematik (volume 1971, issue … Article Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque. was published on January 1, 1971 in the journal Journal für die reine und angewandte Mathematik (volume 1971, issue 247).
Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions. Conventions and notation background material from algebraic geometry general notions associated with algebraic groups homogeneous spaces solvable groups Borel subgroups reductive groups rationality questions.
By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it … By the methods used heretofore for the determination of the automorphisms of certain families of linear groups, for example, the (projective) unimodular, orthogonal, symplectic, and unitary groups (7, 8), it has been necessary to consider the various families separately and to give many case-by-case discussions, especially when the underlying vector space has few elements, even though the final results are very much the same for all of the groups. The purpose of this article is to give a completely uniform treatment of this problem for all the known finite simple linear groups (listed in §2 below). Besides the * ‘classical groups” mentioned above, these include the “exceptional groups,” considered over the complex field by Cartan and over an arbitrary field by Dickson, Chevalley, Hertzig, and the author (3, 4, 5, 6, 10, 15). The automorphisms of the latter groups are given here for the first time.
These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have … These notes are to supplement my paper (4), and should be read in conjunction with it. Both are divided into three parts, and in these notes the section numbers have a further digit added; thus §1.41 here supplements §1.4 of (4). References by section numbers are always to (4) or to the present notes, but references to other papers are numbered independently. The principal results of these notes are the following. New sphere packings are given in [2 m ], m ⩾ 6, and in [24], which are twice as dense as those of §§1.6, 2.3. Others are given in [2 m ], m ⩾ 5, with the same density as those of §1.6, but in which each sphere touches fewer other spheres than in the earlier packings.
This paper is a development of [4], and gives a more detailed treatment of the topic named in the title.It includes in particular the birational equivalence with affine space, over … This paper is a development of [4], and gives a more detailed treatment of the topic named in the title.It includes in particular the birational equivalence with affine space, over the groundfield, of the variety of Cartan subgroups of a k-group G, the splitting of G over a separable extension of k if G is reductive, some results on unipotent groups operated upon by tori, and on the existence of subgroups of G whose Lie algebra contains a given nilpotent element of the Lie algebra g of G.Discussing as it does a number of known results (due mostly to Rosenlicht and Grothendieck), this paper is to be viewed as partly expository.In fact, besides proving some new results, our main goal is to provide a rather comprehensive, albeit not exhaustive, account of our topic, from the point of view sketched in [4],Our basic tools are some rationality properties of transversal intersections and of separable mappings, the Jordan decomposition in g, and purely inseparable isogenies of height one.They are reviewed or discussed in section 1.13, §3 and §5 respectively.Thus Lie algebras of algebraic groups play an important role in this paper and, for the sake of completeness, we have collected in §1 a number of definitions and facts pertaining to them.§2 reproves a result of Grothendieck ([12], Exp.XIV) stating that g is the union of the subalgebras of its Borel subgroups.Its main use for us is to reduce to Lie algebras of solvable groups the existence proof of the Jordan decomposition, §4 discusses subalgebras § of g consisting of semi-simple elements, to be called "toral subalgebras" of g.They are tangent to maximal tori, and have several properties similar to that of tori in G, in particular: the centralizer Z( 8)={g*G 9 Adg(X) = X(X € 8)} of jg in G is defined over k if 3 is, (see 4.3 for Z(3)°, 6.14 for Z(β)\ its Lie algebra is 3(8) = {X z g, [8, X] = 0}.If § is spanned by one element X, the conjugacy class of X is isomorphic to G/Z(β).This paragraph also gives some conditions under which a subalgebra of g is algebraic, and reproves some results of Chevalley [8] in characteristic zero.§6 introduces regular elements, Cartan subalgebras in g, and the subgroups of type (C) of ([12], Exp.XΠI) in G.By definition here, a Cartan subalgebra
In the paper [18], we began a detailed study of the "smallest" group G associated to a Kac-Moody algebra g(A) and of the (in general infinite-dimensional) flag varieties Pν Λ … In the paper [18], we began a detailed study of the "smallest" group G associated to a Kac-Moody algebra g(A) and of the (in general infinite-dimensional) flag varieties Pν Λ associated to G. In the present paper we introduce and study the algebra F[G] of "strongly regular" functions on G. We establish a Peter-Weyl-type decomposition of F[G] with respect to the natural action of G × G (Theorem 1) and prove that F[G] is a unique factorization domain (Theorem 3).