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.

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

Examples
  • For a finite field F, the group of diagonal matrices 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]

See also

References
  1. Springer (1998), § 6.4..
  2. Springer (1998), Proposition 6.4.2. (i).
  3. Lang (2002), p. 712.
  • Borel, Armand (1991-12-31). Linear algebraic groups. ISBN 3-540-97370-2.
  • Lang, Serge (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


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