LieAlgebras[CartanInvolution] - find a Cartan involution for a non-compact, semi-simple, real Lie algebra
Calling Sequences
CartanInvolution(TP)
CartanInvolution( CSARSDPosRts)
Parameters
T - a list of vectors in a Lie algebra, defining a subalgebra on which the Killing form is negative-definite.
P - a list of vectors in a Lie algebra, defining a subspace on which the Killing form is positive-definite.
CSA - a list of vectors, defining a Cartan subalgebra of a Lie algebra
RSD - a table, specifying the root space decomposition of the Lie algebra with respect to the Cartan subalgebra CSA
PosRts - a list of Vectors, specifying a choice of positive roots for the root space decomposition
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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 involution of g is a Lie algebra automorphism Θ : g → g with and such that the symmetric bilinear form is positive-definite.
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The command CartanInvolution returns a transformation defining a Cartan involution.
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A Cartan decomposition is a vector space decomposition g = t ⊕ p , where t is a subalgebra, p a subspace, [t, p] ⊆ p , [p, p] ⊆ t and the Killing form is negative-definite on t and positive-definite on p.
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Given a Cartan decomposition, the linear transformation which is the identity on t and on p is a Cartan involution. This is the involution computed by the first calling sequence for the command CartanInvolution.
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We remark that, conversely, given a Cartan involution, the +1, -1 eigenspaces and yield a Cartan decomposition. Also, any two Cartan involutions and on g are related by an inner automorphism , that is, .
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A Cartan involution can also be calculated from a Cartan subalgebra, the associated root space decomposition and a choice of positive roots. The algorithm can be summarized as follows. First use the procedure Complexify to define the complexification of the Lie algebra . This is a real semi-simple Lie algebra of twice the dimension of . Let denote the standard conjugation map. Next use the command SplitAndCompactForms to find a complex basis of which defines a compact formof . Identify with a subalgebra of and let be the corresponding conjugate map. One proves that is a Cartan involution of . If restricts to a mapping , then would be the required Cartan involution for . However, this generally is not the case so the idea to conjugate to another Cartan involution which does restrict to. Note that the requirement that restricts to a mapping is equivalent to the requirement that commutes with . One proves that is a linear transformation with positive eigenvalues. The required Cartan involution is then . See A.Cap and J. Slovak, Parabolic Geometries I - Background and General Theory, page 203 for further details.
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Examples
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with(DifferentialGeometry): with(LieAlgebras):
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Example 1.
We find a Cartan involution for , the Lie algebra of matrices which are skew-symmetric with respect to the quadratic form .
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LD := SimpleLieAlgebraData("so(3, 2)", so32, labelformat = "gl", labels = ['E', 'omega']):
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The explicit matrices defining are
so32 >
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M := StandardRepresentation(so32);
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From these matrices we calculate a Cartan decomposition
so32 >
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T, P := CartanDecomposition(M, so32);
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and from this a Cartan involution
so32 >
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Theta1 := CartanInvolution(T, P);
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We check that satisfies all the properties of a Cartan involution.
1. .
so32 >
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ComposeTransformations(Theta1, Theta1);
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2. is a Lie algebra homomorphism.
so32 >
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Query(Theta1, "Homomorphism");
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3. The bilinear form is positive-definite.
so32 >
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V := Tools:-DGinfo(so32, "FrameBaseVectors");
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so32 >
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B := Matrix(10, 10, (i,j) -> Killing(-V[i], ApplyHomomorphism(Theta1, V[j])));
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All of these properties are checked with the command Query/"CartanInvolution"
so32 >
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Query(Theta1, "CartanInvolution");
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Example 2.
We calculate the Cartan involution for using the second calling sequence. For this we need a Cartan subalgebra, the corresponding root space decomposition and a choice of positive roots.
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CSA := CartanSubalgebra();
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so32 >
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RSD := RootSpaceDecomposition(CSA);
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so32 >
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PosRts := PositiveRoots(RSD);
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Here is the Cartan involution obtained from this Cartan subalgebra.
so32 >
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Theta2 := CartanInvolution(CSA, RSD, PosRts);
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so32 >
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Query(Theta2, "CartanInvolution");
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It differs slightly from the one calculated using the first calling sequence in Example 1.
Example 3.
We check, by example, that if is an inner automorphism, then is also a Cartan involution.
We use the exponential of to define
so32 >
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A := AdjointExp(evalDG(2*E35));
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so32 >
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phi := Transformation(so32, so32, A);
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Here is the new Cartan involution.
so32 >
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newTheta := ComposeTransformations(phi, Theta1, InverseTransformation(phi));
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Check that it works.
so32 >
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Query(newTheta, "CartanInvolution");
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