Cartan decomposition

Summary

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.[1]

Cartan involutions on Lie algebras

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Let   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.

Examples

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  • A Cartan involution on   is defined by  , where   denotes the transpose matrix of  .
  • The identity map on   is an involution. It is the unique Cartan involution of   if and only if the Killing form of   is negative definite or, equivalently, if and only if   is the Lie algebra of a compact semisimple 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):
    1. The identity involution  , which is the unique Cartan involution in this case.
    2. Complex conjugation, expressible as   on  .
    3. If   is odd,  . The involutions (1), (2) and (3) are equivalent, but not equivalent to the identity involution since  .
    4. If   is even, there is also  .

Cartan pairs

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Let   be an involution on a Lie algebra  . Since  , the linear map   has the two eigenvalues  . If   and   denote the eigenspaces corresponding to +1 and -1, respectively, then  . Since   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

 ,  , and  .

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  , and   is called a symmetric pair. This notion of a Cartan pair here is not to be confused with the distinct notion involving the relative Lie algebra cohomology  .

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  .

Cartan decomposition on the Lie group level

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Let   be a non-compact 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:

  • There is a Lie group automorphism   with differential   at the identity that satisfies  .
  • The subgroup of elements fixed by   is  ; in particular,   is a closed subgroup.
  • The mapping   given by   is a diffeomorphism.
  • The subgroup   is a maximal compact subgroup of  , whenever the center of G is finite.

The automorphism   is also called the global Cartan involution, and the diffeomorphism   is called the global Cartan decomposition. If we write   this says that the product map   is a diffeomorphism so  .

For the general linear group,   is a Cartan involution.[clarification needed]

A refinement of the Cartan decomposition for symmetric spaces of compact or noncompact type states that the maximal Abelian subalgebras   in   are unique up to conjugation by  . Moreover,

 

where  .

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

 

Geometrically the image of the subgroup   in   is a totally geodesic submanifold.

Relation to polar decomposition

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Consider   with the Cartan involution  .[clarification needed] 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. The polar decomposition of an invertible matrix is unique.

See also

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Notes

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References

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  • Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, ISBN 0-8218-2848-7, MR 0514561
  • Kleiner, Israel (2007). Kleiner, Israel (ed.). A History of Abstract Algebra. Boston, MA: Birkhäuser. doi:10.1007/978-0-8176-4685-1. ISBN 978-0817646844. MR 2347309.
  • Knapp, Anthony W. (2005) [1996]. Lie groups beyond an introduction. Progress in Mathematics. Vol. 140 (2nd ed.). Boston, MA: Birkhäuser. ISBN 0-8176-4259-5. MR 1920389.