# Cartan subgroup

In mathematics, a **Cartan subgroup** of a Lie group or algebraic group *G* is one of the subgroups whose Lie algebra is a Cartan subalgebra. The dimension of a Cartan subgroup, and therefore of a Cartan subalgebra, is the **rank** of *G*.

## Conventions

The identity component of a subgroup has the same Lie algebra. There is no *standard* convention for which one of the subgroups with this property is called *the* Cartan subgroup, especially in the case of disconnected groups.

## Definitions

A **Cartan subgroup** of a **compact connected Lie group** is a maximal connected Abelian subgroup (a maximal torus). Its Lie algebra is a Cartan subalgebra.

For **disconnected compact Lie groups** there are several inequivalent definitions of a Cartan subgroup. The most common seems to be the one given by David Vogan, who defines a Cartan subgroup to be the group of elements that normalize a fixed maximal torus and fix the fundamental Weyl chamber. This is sometimes called the **large Cartan subgroup**. There is also a **small Cartan subgroup**, defined to be the centralizer of a maximal torus. These Cartan subgroups need not be abelian in general.

For **connected algebraic groups** over an algebraically closed field a **Cartan subgroup** is usually defined as the centralizer of a maximal torus. In this case the Cartan subgroups are connected, nilpotent, and are all conjugate.

## See also

## References

- Armand Borel (1991-12-31).
*Linear algebraic groups*. ISBN 3-540-97370-2. - Anthony William Knapp; David A. Vogan (1995).
*Cohomological Induction and Unitary Representations*. ISBN 978-0-691-03756-1. - Popov, V. L. (2001) [1994], "C/c020560", in Hazewinkel, Michiel (ed.),
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4