Groupes classiques

Panorama et synthèses, Autour des schémas en groupes vol. II, vol. 46, 2015, p. 1-133
(1^st online appearance of the preprint on the 9-1-2014)

This text contains an extended version of the talks given by the authors at the conference: Schémas en groupes: introdution au séminaire SGA3 de Demazure-Grothendieck, Luminy (Sept. 2011).

We define adjoint and simply connected semi-simple groups of split types A_n, B_n, C_n and D_n over a general base scheme, in “classical terms”, i.e. using Azumaya algebras, quadratic modules etc. In each case, we exhibit the split algebraic group (Chevalley group) and its root datum, and then all (étale or fppf) twisted forms, using concrete descriptions of categories of torsors and the contracted product. We do not do the outer D₄ case nor exceptional types.