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 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, § 6.4.
  2. Springer, Proposition 6.4.2. (i)
  3. Serge Lang (2002). Algebra. p. 712.


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