In mathematics, a quasi-split group over a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

is a

reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...

with a

Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...

defined over the field. Simply connected quasi-split groups over a field correspond to actions of the absolute

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...

on a

Dynkin diagram In the Mathematics, mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of Graph (discrete mathematics), graph with some edges doubled or tripled (drawn as a double or triple line). Dynkin diagrams arise in the ...

Examples

All

split group Split(s) or The Split may refer to: Places * Split, Croatia, the largest coastal city in Croatia * Split Island, Canada, an island in the Hudson Bay * Split Island, Falkland Islands * Split Island, Fiji, better known as Hạfliua Arts, entert ...

s (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. 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 ''O''_{''n'',''n''+2}, the unitary groups ''SU''_''n'',''n'' and ''SU''_{''n'',''n''+1}, and the form of ''E''₆ with signature 2.

References

*{{Citation , last1=Lang , first1=Serge , author1-link=Serge Lang , title=Algebraic groups over finite fields , jstor=2372673 , mr=0086367 , year=1956 , journal=

American Journal of Mathematics The ''American Journal of Mathematics'' is a bimonthly mathematics journal published by the Johns Hopkins University Press. History The ''American Journal of Mathematics'' is the oldest continuously published mathematical journal in the United S ...

, issn=0002-9327 , volume=78 , pages=555–563 , doi=10.2307/2372673 Linear algebraic groups