Cartan subgroup

For a Cartan subgroup of a Lie group, see Cartan subalgebra § Cartan subgroup.

In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group $G$ over a (not necessarily algebraically closed) field $k$ is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If $k$ is algebraically closed, they are all conjugate to each other.

Notice that in the context of algebraic groups a torus is an algebraic group $T$ such that the base extension $T_{({\bar {k}})}$ (where ${\bar {k}}$ is the algebraic closure of $k$) is isomorphic to the product of a finite number of copies of the $\mathbf {G} _{m}=\mathbf {GL} _{1}$. 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 $G$ is reductive (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser and thus Cartan subgroups of $G$ are precisely the maximal tori.

Example

The general linear groups $\mathbf {GL} _{n}$ are reductive. The diagonal subgroup is clearly a torus (indeed a split torus, since it is product of n copies of $\mathbf {G} _{m}$ already before any base extension), and it can be shown to be maximal. Since $\mathbf {GL} _{n}$ is reductive, the diagonal subgroup is a Cartan subgroup.

See also

• Borel subgroup
• Algebraic group
• Algebraic torus

References

• 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