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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. Early life and education Tits was born in Uccle, Belgium 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 , and Tits graduated with his doctorate in 1950 with the dissertation ''Généralisation des groupes projectifs basés sur la notion de transitivité''. Career Tits held 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 Collège de F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uccle
Uccle (French language, French, ) or Ukkel (Dutch language, Dutch, ) is one of the List of municipalities of the Brussels-Capital Region, 19 municipalities of the Brussels-Capital Region, Belgium. Located in the southern part of the region, it is bordered by the City of Brussels, Forest, Belgium, Forest, Ixelles, and Watermael-Boitsfort, as well as the Flanders, Flemish municipalities of Drogenbos, Linkebeek and Sint-Genesius-Rode. In common with all of Brussels' municipalities, it is legally Multilingualism, bilingual (French–Dutch). , the municipality had a population of 85,099 inhabitants. The total area is , which gives a population density of , half the average of Brussels. It is generally considered an affluent area of the region, and is particularly noted for its community of French immigrants. History Origins and medieval times 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 Liè ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tits Alternative
In mathematics, the Tits alternative, named after 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 finite ... [...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 ''Ruth Moufang, 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''- ... [...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 operations, 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 theories, there is a field-like object whose characteristic is one. Most proposed theories of F1 replace abstract algebra entirely. Mathematical objects such as vector spaces and polynomial rings can be carried over into these new theories by mimickin ... [...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. Tannaka-Artin problem">problème de Tannaka-Artin, M.Kneser a posé la question suivante que j’ai imprudemment transformé en conjecture.''" - J. Tits 1978.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 metric induced by a discrete valuatio ... [...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 In abstract algebra, a Jordan algebra is a nonassociative algebra over a field whose multiplication satisfies the following axioms: # xy = yx (commutative law) # (xy)(xx) = x(y(xx)) (). The product of two elements ''x'' and ''y'' in a Jordan a ..., 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 27-dimensional exceptional Jordan algebra it gives a Lie algebra of type 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 the 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 ... [...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 ... [...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 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 ... [...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] |
