Langbahn Team – Weltmeisterschaft

Cartan subgroup

In the theory of algebraic groups, a Cartan subgroup of a connected linear algebraic group over a (not necessarily algebraically closed) field is the centralizer of a maximal torus. Cartan subgroups are smooth (equivalently reduced), connected and nilpotent. If is algebraically closed, they are all conjugate to each other. [1]

Notice that in the context of algebraic groups a torus is an algebraic group such that the base extension (where is the algebraic closure of ) is isomorphic to the product of a finite number of copies of the . 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 is reductive (in particular, if it is semi-simple), then a torus is maximal if and only if it is its own centraliser [2] and thus Cartan subgroups of are precisely the maximal tori.

Example

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

See also

References

  1. ^ Milne (2017), Proposition 17.44.
  2. ^ Milne (2017), Corollary 17.84.