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Concept
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.
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 . 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.
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