
Cartan decomposition
Encyclopedia
The Cartan decomposition is a decomposition of a semisimple Lie group
or Lie algebra
, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition of matrices.
be a real semisimple Lie algebra
and let
be its Killing form
. An involution on
is a Lie algebra automorphism
of
whose square is equal to the identity. Such an involution is called a Cartan involution on
if
is a positive definite bilinear form.
Two involutions
and
are considered equivalent if they differ only by an inner automorphism.
Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equivalent.
be an involution on a Lie algebra
. Since
, the linear map
has the two eigenvalues
. Let
and
be the corresponding eigenspaces, then
. Since
is a Lie algebra automorphism, eigenvalues are multiplicative. It follows that
Thus
is a Lie subalgebra, while any subalgebra of
is commutative.
Conversely, a decomposition
with these extra properties determines an involution
on
that is
on
and
on
.
Such a pair
is also called a Cartan pair of
.
The decomposition
associated to a Cartan involution is called a Cartan decomposition of
. The special feature of a Cartan decomposition is that the Killing form is negative definite on
and positive definite on
. Furthermore,
and
are orthogonal complements of each other with respect to the Killing form on
.
be a semisimple Lie group
and
its Lie algebra
. Let
be a Cartan involution on
and let
be the resulting Cartan pair. Let
be the analytic subgroup of
with Lie algebra
. Then
The automorphism
is also called global Cartan involution, and the diffeomorphism
is called global Cartan decomposition.
For the general linear group, we get
as the Cartan involution.
with the Cartan involution
. Then
is the real Lie algebra of skew-symmetric matrices, so that
, while
is the subspace of symmetric matrices. Thus the exponential map is a diffeomorphism from
onto the space of positive definite matrices. Up to this exponential map, the global Cartan decomposition is the polar decomposition of a matrix. Notice that the polar decomposition of an invertible matrix is unique.
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
or Lie algebra
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
, which plays an important role in their structure theory and representation theory. It generalizes the polar decomposition of matrices.
Cartan involutions on Lie algebras
Let
Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
and let

Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras...
. An involution on

Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...




Two involutions


Any real semisimple Lie algebra has a Cartan involution, and any two Cartan involutions are equivalent.
Examples
- A Cartan involution on
is defined by
, where
denotes the transpose matrix of
.
- The identity map on
is an involution, of course. It is the unique Cartan involution of
if and only if the Killing form of
is negative definite. Equivalently,
is the Lie algebra of a compact Lie group.
- Let
be the complexification of a real semisimple Lie algebra
, then complex conjugation on
is an involution on
. This is the Cartan involution on
if and only if
is the Lie algebra of a compact Lie group.
- The following maps are involutions of the Lie algebra
of the special unitary group SU(n):
-
- the identity involution
, which is the unique Cartan involution in this case;
- the identity involution
-
-
which on
is also the complex conjugation;
-
-
- if
is odd,
. These are all equivalent, but not equivalent to the identity involution (because the matrix
does not belong to
.)
- if
-
- if
is even, we also have
- if
Cartan pairs
Let








-
,
, and
.
Thus


Conversely, a decomposition







Such a pair


The decomposition







Cartan decomposition on the Lie group level
Let
Lie group
In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
and

Lie algebra
In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
. Let






- There is a Lie group automorphism
with differential
that satisfies
.
- The subgroup of elements fixed by
is
; in particular,
is a closed subgroup.
- The mapping
given by
is a diffeomorphism.
- The subgroup
contains the center
of
, and
is compact modulo center, that is,
is compact.
- The subgroup
is the maximal subgroup of
that contains the center and is compact modulo center.
The automorphism


For the general linear group, we get

Relation to polar decomposition
Consider




