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).^[1] Cartan subgroups are nilpotent^[2] 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).^[3]
For a finite field F, every maximal commutative semisimple subgroup of GL₂(F) is a Cartan subgroup (and conversely).^[3]

References

^ Springer, § 6.4.
^ Springer, Proposition 6.4.2. (i)
^ ^a ^b Serge Lang (2002). Algebra. p. 712.

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, 9 (2nd ed.), Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4021-7, MR 1642713

[1] Springer, § 6.4.

[2] Springer, Proposition 6.4.2. (i)

[Lang-3] Serge Lang (2002). Algebra. p. 712.

[1]

[2]

[3]

Cartan subgroup

Examples

See also

References