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
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.
- 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):
- The identity involution , which is the unique Cartan involution in this case.
- Complex conjugation, expressible as on .
- If is odd, . The involutions (1), (2) and (3) are equivalent, but not equivalent to the identity involution since .
- If is even, there is also .
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, eigenvalues are multiplicative. 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
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 contains the center of , and is compact.
- The subgroup is the maximal subgroup of that contains the center and for which is compact.
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.
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,
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
Consider 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. The polar decomposition of an invertible matrix is unique.
- Helgason, Sigurdur (1978), Differential geometry, Lie groups, and symmetric spaces, Academic Press, ISBN 0-8218-2848-7
- Kleiner, I. (2007). A History of Abstract Algebra. Birkhäuser. ISBN 978-0817646844.
- Knapp, A. W. (2005) . Bass, Hyman; Oesterlé, Joseph; Alan, Weinstein (eds.). Lie groups beyond an introduction. Progress in mathematics. 140 (2nd ed.). Birkhäuser. ISBN 0-8176-4259-5.