In mathematics, the Kneser–Tits problem, introduced by Tits (1964) 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. ["Généralisant le 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
editA 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. Platonov (1969) showed that this Whitehead group is trivial for local fields K, and gave examples of fields for which it is not always trivial. For global fields the combined work of several authors shows that this Whitehead group is always trivial (Gille 2009).
References
edit- Gille, Philippe (2009), "Le problème de Kneser-Tits" (PDF), Astérisque, Séminaire Bourbaki exp. 983 (326): 39–81, ISBN 978-2-85629-269-3, ISSN 0303-1179, MR 2605318
- Platonov, V. P. (1969), "The problem of strong approximation and the Kneser–Tits hypothesis for algebraic groups", Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, 33: 1211–1219, ISSN 0373-2436, MR 0258839
- Tits, Jacques (1978), "Groupes de Whitehead de groupes algébriques simples sur un corps (d'après V. P. Platonov et al.)", Séminaire Bourbaki, 29e année (1976/77), Lecture Notes in Math., vol. 677, Berlin, New York: Springer-Verlag, pp. 218–236, MR 0521771
- Tits, Jacques (1964), "Algebraic and abstract simple groups", Annals of Mathematics, Second Series, 80: 313–329, doi:10.2307/1970394, ISSN 0003-486X, JSTOR 1970394, MR 0164968
External links
edit- "Kneser-Tits hypothesis", Encyclopedia of Mathematics, EMS Press, 2001 [1994]