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Wikipedia / Cartan decomposition

In mathematics, 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 or singular value decomposition of matrices. Its history can be traced to the 1880s work of Élie Cartan and Wilhelm Killing.

Cartan involutions on Lie algebras

be a real semisimple Lie algebra be its Killing form. An involution is a Lie algebra automorphism whose square is equal to the identity. Such an involution is called a Cartan involution is a positive definite bilinear form.

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.

Examples

  • .

  • is the Lie algebra of a compact semisimple Lie group.

  • be the complexification is the Lie algebra of a compact Lie group.

of the special unitary group SU(n):

  1. , which is the unique Cartan involution in this case.

  2. Complex conjugation.

  3. .

  4. .

Cartan pairs

is a Lie algebra automorphism, the Lie bracket of two of its eigenspaces is contained in the eigenspace corresponding to the product of their eigenvalues. It follows that

  1. .

is commutative.

.

is also called a Cartan pair is called a symmetric pair. This notion of a Cartan pair here is not to be confused with the distinct notion.

associated to a Cartan involution is called a Cartan decomposition.

Cartan decomposition on the Lie group level

be the analytic subgroup. Then:

  • .

  • is a closed subgroup.

  • is a diffeomorphism.

  • , whenever the center of G is finite.

is also called the global Cartan involution is called the global Cartan decomposition.

For the general linear group is a Cartan involution.

. Moreover,

.

In the compact and noncompact case the global Cartan decomposition thus implies

is a totally geodesic submanifold.

Relation to polar decomposition

. onto the space of positive definite matrices. Up to this exponential map, the global Cartan decomposition is the polar decomposition of a matrix. The polar decomposition of an invertible matrix is unique.

See also

Notes

References