Quasi-split group
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In mathematics, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. Simply connected quasi-split groups over a field correspond to actions of the absolute Galois group on a Dynkin diagram.
Examples
[edit]All split groups (those with a split maximal torus) are quasi-split. These correspond to quasi-split groups where the action of the Galois group on the Dynkin diagram is trivial.
Lang (1956) showed that all simple algebraic groups over finite fields are quasi-split.
Over the real numbers, the quasi-split groups include the split groups and the complex groups, together with the orthogonal groups On,n+2, the unitary groups SUn,n and SUn,n+1, and the form of E6 with signature 2.
References
[edit]- Lang, Serge (1956), "Algebraic groups over finite fields", American Journal of Mathematics, 78: 555–563, doi:10.2307/2372673, ISSN 0002-9327, JSTOR 2372673, MR 0086367