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LieAlgebras[CartanDecomposition] - find the Cartan decomposition defined by a Cartan involution, find the Cartan decomposition of a semi-simple matrix algebra
Calling Sequences
CartanDecomposition( )
CartanDecomposition( )
Parameters
Theta - a transformation, defining a Cartan involution of a non-compact, semi-simple real Lie algebra
 A - a list of square matrices, defining a Lie algebra and closed under Hermitian tranposition
 alg - a name or a string, the name of an initialized Lie algebra
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Description
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Let g be a semi-simple real Lie algebra. Then g is called compact if the Killing form of g is negative-definite, otherwise g is called non-compact.
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A Cartan decomposition is a vector space decomposition g = t 4p , where t is a subalgebra, p a subspace, [t, p] 4 p and [p, p] 4 t, the Killing form  is negative-definite on t and positive-definite on p.
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For a semi-simple matrix algebra which is closed under Hermitian transposition, the decomposition into skew-Hermitian and Hermitian matrices will give a Cartan decomposition.
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Examples
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with(DifferentialGeometry): with(LieAlgebras):
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Example 1.
We find a Cartan decomposition for the Lie algebra  from a Cartan involution.
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LD := SimpleLieAlgebraData("sl(3)", sl3, labelformat = "gl", labels = ['E', 'omega']):
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| (2.1) |
The transformation  defines a Cartan involution for  . With respect to the standard basis for  in terms of elementary matrices, this transformation is given by
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sl3 >
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Theta := Transformation([[E11, -E11], [E22, -E22], [E12, -E21], [E13, -E31], [E21, -E12], [E23, -E32], [E31, -E13], [E32, -E23]]);
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![_DG([["transformation", [[sl3, 0], [sl3, 0]], [], [Matrix(%id = 102171796)]], [[-_z1, _z1], [-_z2, _z2], [-_z5, _z3], [-_z7, _z4], [-_z3, _z5], [-_z8, _z6], [-_z4, _z7], [-_z6, _z8]]])](/support/helpjp/helpview.aspx?si=6584/file05784/math110.png)
| (2.2) |
The corresponding Cartan decomposition is given by
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sl3 >
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T, P := CartanDecomposition(Theta);
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![[_DG([["vector", sl3, []], [[[6], -1], [[8], 1]]]), _DG([["vector", sl3, []], [[[4], -1], [[7], 1]]]), _DG([["vector", sl3, []], [[[3], -1], [[5], 1]]])], [_DG([["vector", sl3, []], [[[6], 1], [[8], 1]]]), _DG([["vector", sl3, []], [[[4], 1], [[7], 1]]]), _DG([["vector", sl3, []], [[[3], 1], [[5], 1]]]), _DG([["vector", sl3, []], [[[2], 1]]]), _DG([["vector", sl3, []], [[[1], 1]]])]](/support/helpjp/helpview.aspx?si=6584/file05784/math121.png)
| (2.3) |
Let us check the various properties of this decomposition.
1.  is a subalgebra.
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sl3 >
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Query(T, "Subalgebra");
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| (2.4) |
2. [![T, P] subset P](/support/helpjp/helpview.aspx?si=6584/file05784/math144.png)
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sl3 >
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A := BracketOfSubspaces(T, P);
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![[_DG([["vector", sl3, []], [[[2], -2]]]), _DG([["vector", sl3, []], [[[3], -1], [[5], -1]]]), _DG([["vector", sl3, []], [[[4], 1], [[7], 1]]]), _DG([["vector", sl3, []], [[[6], 2], [[8], 2]]]), _DG([["vector", sl3, []], [[[1], -2], [[2], -2]]])]](/support/helpjp/helpview.aspx?si=6584/file05784/math151.png)
| (2.5) |
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sl3 >
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GetComponents(A, P, trueorfalse = "on");
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| (2.6) |
3. [![P, P] subset T](/support/helpjp/helpview.aspx?si=6584/file05784/math163.png)
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sl3 >
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B := BracketOfSubspaces(P,P);
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![[_DG([["vector", sl3, []], [[[3], -1], [[5], 1]]]), _DG([["vector", sl3, []], [[[4], -1], [[7], 1]]]), _DG([["vector", sl3, []], [[[6], -2], [[8], 2]]])]](/support/helpjp/helpview.aspx?si=6584/file05784/math171.png)
| (2.7) |
4. Equivalent to 1, 2 and 3 is the fact that  form a symmetric pair.
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sl3 >
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Query(T, P, "SymmetricPair");
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5. The Killing form is negative-definite on T.
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![Matrix([[-12, 0, 0], [0, -12, 0], [0, 0, -12]])](/support/helpjp/helpview.aspx?si=6584/file05784/math198.png)
| (2.9) |
6. The Killing form is positive-definite on P.
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| (2.10) |
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sl3 >
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LinearAlgebra:-IsDefinite(KP);
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| (2.11) |
All of these properties of the Cartan decomposition can be checked at once with the Query/"CartanDecomposition" command.
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sl3 >
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Query(T, P, "CartanDecomposition");
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| (2.12) |
Example 2.
Here we shall calculate the Cartan decomposition of  from its standard matrix representation. We use the Lie algebra initialized in Example 1. The command StandardRepresentation can be applied to any Lie algebra created by  .
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sl3 >
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M := StandardRepresentation(sl3);
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| (2.13) |
Calculate the Cartan decomposition in terms of matrices.
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sl3 >
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T2, P2 := CartanDecomposition(M);
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| (2.14) |
Note that the matrices in  are skew-symmetric and that the matrices in  are symmetric.
Calculate the Cartan decomposition using the matrices (2.13) but return the answer as vectors in the Lie algebra 
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sl3 >
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T3, P3 := CartanDecomposition(M, sl3);
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![[_DG([["vector", sl3, []], [[[3], 1], [[5], -1]]]), _DG([["vector", sl3, []], [[[4], 1], [[7], -1]]]), _DG([["vector", sl3, []], [[[6], 1], [[8], -1]]])], [_DG([["vector", sl3, []], [[[1], 2]]]), _DG([["vector", sl3, []], [[[2], 2]]]), _DG([["vector", sl3, []], [[[3], 1], [[5], 1]]]), _DG([["vector", sl3, []], [[[4], 1], [[7], 1]]]), _DG([["vector", sl3, []], [[[6], 1], [[8], 1]]])]](/support/helpjp/helpview.aspx?si=6584/file05784/math282.png)
| (2.15) |
Example 3.
Here we shall calculate the Cartan decomposition of  from its standard matrix representation.
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LD := SimpleLieAlgebraData("sp(4, 2)", sp42, labelformat = "gl", labels = ['F', 'sigma']):
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| (2.16) |
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sl3 >
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M := StandardRepresentation(sp42);
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| (2.17) |
Here is a Cartan decomposition for  .
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sp42 >
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T, P := CartanDecomposition(M, sp42);
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| (2.18) |
Check it.
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sp42 >
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Query(T, P, "CartanDecomposition");
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| (2.19) |
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