In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group over a (not necessarily algebraically closed) field is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If is algebraically closed, they are all conjugate to each other. [1]
Notice that in the context of algebraic groups a torus is an algebraic group such that the base extension (where is the algebraic closure of ) is isomorphic to the product of a finite number of copies of the . Maximal such subgroups have in the theory of algebraic groups a role that is similar to that of maximal tori in the theory of Lie groups.
If is reductive (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser [2] and thus Cartan subgroups of are precisely the maximal tori.
Example[edit]
The general linear groups are reductive. The diagonal subgroup is clearly a torus (indeed a split torus, since it is product of n copies of already before any base extension), and it can be shown to be maximal. Since is reductive, the diagonal subgroup is a Cartan subgroup.
See also[edit]
References[edit]
- ^ Milne (2017), Proposition 17.44.
- ^ Milne (2017), Corollary 17.84.
- Borel, Armand (1991-12-31). Linear algebraic groups. ISBN 3-540-97370-2.
- Lang, Serge (2002). Algebra. Springer. ISBN 978-0-387-95385-4.
- Milne, J. S. (2017), Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press, doi:10.1017/9781316711736, ISBN 978-1107167483, MR 3729270
- Popov, V. L. (2001) [1994], "Cartan subgroup", Encyclopedia of Mathematics, EMS Press
- Springer, Tonny A. (1998), Linear algebraic groups, Progress in Mathematics, vol. 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713