Cartan subgroup

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

In algebraic geometry, a Cartan subgroup of a connected linear algebraic group over an algebraically closed field is the centralizer of a maximal torus (which turns out to be connected). Cartan subgroups are nilpotent and are all conjugate.

Examples

• For a finite field F, the group of diagonal matrices ${\begin{pmatrix}a&0\\0&b\end{pmatrix}}$ where a and b are elements of F^*. This is called the split Cartan subgroup of GL₂(F).
• For a finite field F, every maximal commutative semisimple subgroup of GL₂(F) is a Cartan subgroup (and conversely).

See also

• Borel subgroup

References

• Armand Borel (1991-12-31). Linear algebraic groups. ISBN 3-540-97370-2.
• Serge Lang (2002). Algebra. Springer. ISBN 978-0-387-95385-4.
• 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