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Jacques Tits
Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life and career Tits was born in Uccle to Léon Tits, a professor, and Lousia André. Jacques attended the Athénée of Uccle and the Free University of Brussels (1834–1969), Free University of Brussels. His thesis advisor was Paul Libois, and Tits graduated with his doctorate in 1950 with the dissertation ''Généralisation des groupes projectifs basés sur la notion de transitivité''. His academic career includes professorships at the Free University of Brussels (now split into the Université Libre de Bruxelles and the Vrije Universiteit Brussel) (1962–1964), the University of Bonn (1964–1974) and the Collège de France in Paris, until becoming emeritus in 2000. He changed his citizenship to French in 1974 in order to teach at the Collèg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uccle
Uccle () or Ukkel () is one of the 19 municipalities of the Brussels-Capital Region, Belgium. In common with all of Brussels' municipalities, it is legally bilingual (French–Dutch). It is generally considered an affluent area of the city and is particularly noted for its community of French immigrants. History According to legend, Uccle's church of St. Peter was dedicated by Pope Leo III in the year 803, with Charlemagne and Gerbald, Bishop of Liège, attending the ceremony. During the following centuries, several noble families built their manors and took residency there. The first mention of the name ''Woluesdal'', now evolved into ''Wolvendael'', dates from 1209. In 1467, Isabella of Portugal, wife of Philip the Good, Duke of Burgundy, founded a Franciscan convent on Uccle's territory. Later, Uccle became the judiciary capital of the area including Brussels. Throughout the early stages of its history, however, the village of Uccle always had a predominantly rural char ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Alternative
In mathematics, the Tits alternative, named for Jacques Tits, is an important theorem about the structure of finitely generated linear groups. Statement The theorem, proven by Tits, is stated as follows. Consequences A linear group is not amenable if and only if it contains a non-abelian free group (thus the von Neumann conjecture, while not true in general, holds for linear groups). The Tits alternative is an important ingredient in the proof of Gromov's theorem on groups of polynomial growth. In fact the alternative essentially establishes the result for linear groups (it reduces it to the case of solvable groups, which can be dealt with by elementary means). Generalizations In geometric group theory, a group ''G'' is said to satisfy the Tits alternative if for every subgroup ''H'' of ''G'' either ''H'' is virtually solvable or ''H'' contains a nonabelian free subgroup (in some versions of the definition this condition is only required to be satisfied for all finit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Theory
In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field (mathematics), fields, and vector spaces, can all be seen as groups endowed with additional operation (mathematics), operations and axioms. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. Linear algebraic groups and Lie groups are two branches of group theory that have experienced advances and have become subject areas in their own right. Various physical systems, such as crystals and the hydrogen atom, and Standard Model, three of the four known fundamental forces in the universe, may be modelled by symmetry groups. Thus group theory and the closely related representation theory have many important applications in physics, chemistry, and materials science. Group theory is also ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Polygon
In mathematics, a generalized polygon is an incidence structure introduced by Jacques Tits in 1959. Generalized ''n''-gons encompass as special cases projective planes (generalized triangles, ''n'' = 3) and generalized quadrangles (''n'' = 4). Many generalized polygons arise from groups of Lie type, but there are also exotic ones that cannot be obtained in this way. Generalized polygons satisfying a technical condition known as the '' Moufang property'' have been completely classified by Tits and Weiss. Every generalized ''n''-gon with ''n'' even is also a near polygon. Definition A generalized ''2''-gon (or a digon) is an incidence structure with at least 2 points and 2 lines where each point is incident to each line. For ''n \geq 3'' a generalized ''n''-gon is an incidence structure (P,L,I), where P is the set of points, L is the set of lines and I\subseteq P\times L is the incidence relation, such that: * It is a partial linear space. * It has no ordinary ''m''-gons as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field With One Element
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name "field with one element" and the notation F1 are only suggestive, as there is no field with one element in classical abstract algebra. Instead, F1 refers to the idea that there should be a way to replace sets and operation Operation or Operations may refer to: Arts, entertainment and media * ''Operation'' (game), a battery-operated board game that challenges dexterity * Operation (music), a term used in musical set theory * ''Operations'' (magazine), Multi-Man ...s, the traditional building blocks for abstract algebra, with other, more flexible objects. Many theories of F1 have been proposed, but it is not clear which, if any, of them give F1 all the desired properties. While there is still no field with a single element in these ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kneser–Tits Conjecture
In mathematics, the Kneser–Tits problem, introduced by based on a suggestion by Martin Kneser, asks whether the Whitehead group ''W''(''G'',''K'') of a semisimple simply connected isotropic algebraic group ''G'' over a field ''K'' is trivial. The Whitehead group is the quotient of the rational points of ''G'' by the normal subgroup generated by ''K''-subgroups isomorphic to the additive group. Fields for which the Whitehead group vanishes A special case of the Kneser–Tits problem asks for which fields the Whitehead group of a semisimple almost simple simply connected isotropic algebraic group is always trivial. showed that this Whitehead group is trivial for local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compa ...s ''K'', and gave examples of fields for which it is no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kantor–Koecher–Tits Construction
In algebra, the Kantor–Koecher–Tits construction is a method of constructing a Lie algebra from a Jordan algebra, introduced by , , and . If ''J'' is a Jordan algebra, the Kantor–Koecher–Tits construction puts a Lie algebra structure on ''J'' + ''J'' + Inner(''J''), the sum of 2 copies of ''J'' and the Lie algebra of inner derivations of ''J''. When applied to a Albert algebra, 27-dimensional exceptional Jordan algebra it gives a Lie algebra of type E7 (mathematics), E7 of dimension 133. The Kantor–Koecher–Tits construction was used by to classify the finite-dimensional simple Jordan superalgebras. References * * * * * {{DEFAULTSORT:Kantor-Koecher-Tits construction Lie algebras Non-associative algebras ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Freudenthal–Tits Magic Square
In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups). It is named after Hans Freudenthal and Jacques Tits, who developed the idea independently. It associates a Lie algebra to a pair of division algebras ''A'', ''B''. The resulting Lie algebras have Dynkin diagrams according to the table at right. The "magic" of the Freudenthal magic square is that the constructed Lie algebra is symmetric in ''A'' and ''B'', despite the original construction not being symmetric, though Vinberg's symmetric method gives a symmetric construction. The Freudenthal magic square includes all of the exceptional Lie groups apart from ''G''2, and it provides one possible approach to justify the assertion that "the exceptional Lie groups all exist because of the octonions": ''G''2 itself is the automorphism group of the octonions (also, it is in many ways like a classical Lie group because it is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Space
In geometry, an Hadamard space, named after Jacques Hadamard, is a non-linear generalization of a Hilbert space. In the literature they are also equivalently defined as complete CAT(0) spaces. A Hadamard space is defined to be a nonempty complete metric space such that, given any points x and y, there exists a point m such that for every point z, d(z, m)^2 + \leq . The point m is then the midpoint of x and y: d(x, m) = d(y, m) = d(x, y)/2. In a Hilbert space, the above inequality is equality (with m = (x+y)/2), and in general an Hadamard space is said to be if the above inequality is equality. A flat Hadamard space is isomorphic to a closed convex subset of a Hilbert space. In particular, a normed space is an Hadamard space if and only if it is a Hilbert space. The geometry of Hadamard spaces resembles that of Hilbert spaces, making it a natural setting for the study of rigidity theorems. In a Hadamard space, any two points can be joined by a unique geodesic between them; in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
(B, N) Pair
In mathematics, a (''B'', ''N'') pair is a structure on groups of Lie type that allows one to give uniform proofs of many results, instead of giving a large number of case-by-case proofs. Roughly speaking, it shows that all such groups are similar to the general linear group over a field. They were introduced by the mathematician Jacques Tits, and are also sometimes known as Tits systems. Definition A (''B'', ''N'') pair is a pair of subgroups ''B'' and ''N'' of a group ''G'' such that the following axioms hold: * ''G'' is generated by ''B'' and ''N''. * The intersection, ''T'', of ''B'' and ''N'' is a normal subgroup of ''N''. *The group ''W'' = ''N/T'' is generated by a set ''S'' of elements of order 2 such that **If ''s'' is an element of ''S'' and ''w'' is an element of ''W'' then ''sBw'' is contained in the union of ''BswB'' and ''BwB''. **No element of ''S'' normalizes ''B''. The set ''S'' is uniquely determined by ''B'' and ''N'' and the pair (''W'',''S'') is a Coxeter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Metric
In mathematics, the Tits metric is a metric defined on the ideal boundary of an Hadamard space (also called a complete CAT(0) space). It is named after Jacques Tits. Ideal boundary of an Hadamard space Let (''X'', ''d'') be an Hadamard space. Two geodesic rays ''c''1, ''c''2 : , ∞→ ''X'' are called asymptotic if they stay within a certain distance when traveling, i.e. :\sup_ d(c_1(t), c_2(t)) < \infty. Equivalently, the Hausdorff distance between the two rays is finite. The asymptotic property defines an on the set of geodesic rays, and the set of equivalence classes is called the ideal boundary ∂''X'' of ''X''. An equivalence class of geodesic rays is called a boundary point of ''X''. For any equivalence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
List Of Irreducible Tits Indices
In the mathematical theory of linear algebraic groups, a Tits index (or index) is an object used to classify semisimple algebraic groups defined over a base field ''k'', not assumed to be algebraically closed. The possible irreducible indices were classified by Jacques Tits, and this classification is reproduced below. (Because every index is a direct sum of irreducible indices, classifying ''all'' indices amounts to classifying irreducible indices.) Organization of the list An index can be represented as a Dynkin diagram with certain vertices drawn close to each other (the orbit of the vertices under the *-action of the Galois group of ''k'') and with certain sets of vertices circled (the orbits of the non-distinguished vertices under the *-action). This representation captures the full information of the index except when the underlying Dynkin diagram is D4, in which case one must distinguish between an action by the cyclic group ''C''3 or the permutation group ''S''3. Alternati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
