Chevalley groups
E559863
Chevalley groups are a broad class of linear algebraic groups constructed over arbitrary fields that generalize classical Lie groups and play a central role in the classification of finite simple groups.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic group constructions
ⓘ
class of groups ⓘ linear algebraic groups ⓘ |
| admit |
BN-pairs
ⓘ
Bruhat decomposition ⓘ |
| are |
connected linear algebraic groups
ⓘ
reductive groups ⓘ |
| areCharacterizedBy | root datum ⓘ |
| areDefinedOver |
algebraically closed fields
ⓘ
arbitrary fields ⓘ finite fields ⓘ local fields ⓘ number fields ⓘ |
| areOften | simple as algebraic groups ⓘ |
| areRelatedTo |
Tits systems
ⓘ
buildings in the sense of Tits ⓘ |
| areSometimesExtendedTo |
Steinberg groups
NERFINISHED
ⓘ
twisted groups of Lie type ⓘ |
| centralRoleIn | classification of finite simple groups ⓘ |
| constructedFrom |
root systems
ⓘ
semisimple Lie algebras ⓘ |
| constructedUsing | Chevalley basis NERFINISHED ⓘ |
| fieldOfStudy |
Lie theory
ⓘ
algebraic geometry ⓘ group theory ⓘ representation theory ⓘ |
| generalize | classical Lie groups ⓘ |
| have |
Borel subgroups
NERFINISHED
ⓘ
maximal tori ⓘ unipotent radicals ⓘ |
| haveSubtype |
adjoint Chevalley groups
ⓘ
finite Chevalley groups ⓘ simply connected Chevalley groups NERFINISHED ⓘ split simple algebraic groups ⓘ |
| include |
exceptional algebraic groups
ⓘ
groups of type E6 ⓘ groups of type E7 ⓘ groups of type E8 ⓘ groups of type F4 ⓘ groups of type G2 ⓘ orthogonal groups ⓘ special linear groups ⓘ spin groups ⓘ symplectic groups ⓘ |
| introducedIn | 1950s ⓘ |
| namedAfter | Claude Chevalley NERFINISHED ⓘ |
| usedIn | construction of finite groups of Lie type ⓘ |
| wereIntroducedBy | Claude Chevalley NERFINISHED ⓘ |
| yield | many finite simple groups ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.