- Cartan decomposition
The Cartan decomposition is a decomposition of a semisimple
Lie group orLie algebra , which plays an important role in their structure theory and representation theory. It generalizes thepolar decomposition of matrices.Cartan involutions on Lie algebras
Let be a real
semisimple Lie algebra and let be itsKilling form . An involution on is a Lie algebraautomorphism of whose square is equal to the identity automorphism. Such an involution is called a Cartan involution on if is apositive 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
* 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;
** ;
** , where ; these are not equivalent to the identity involution because the matrix does not belong to .
** if is even, we also have
Cartan pairs
Let 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, we have: , , and .Thus is a Lie subalgebra, while is not.
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 Cartain involution is called a Cartan decomposition of . The special feature of a Cartan decomposition is that the Kiling 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
Let be a
semisimple Lie group and itsLie algebra . Let be a Cartan involution on and let be the resulting Cartan pair. Let be theanalytic subgroup of with Lie algebra . Then
* 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 is also called global Cartan involution, and the diffeomorphism is called global Cartan decomposition.
Relation to polar decompostion
Consider with the Cartain involution . Then is the Lie algebra of skew-symmetric matrices, so that , while is the subspace of positive definite 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.See also
*
Lie group decompositions References
*
A. W. Knapp , "Lie groups beyond an introduction", ISBN 0-8176-4259-5, Birkhäuser.
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