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

• Cartan subalgebra
• Carter subgroup

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), "C/c020560", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4