LieAlgebras[CartanInvolution] - find a Cartan involution for a non-compact, semi-simple, real Lie algebra
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
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.
A Cartan involution of g is a Lie algebra automorphism Θ : g → g with Θ2= Id and such that the symmetric bilinear form BΘx,y = −x,Θy is positive-definite.
The command CartanInvolution returns a transformation defining a Cartan involution.
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.
Given a Cartan decomposition, the linear transformation which is the identity Id on t and −Id on p is a Cartan involution. This is the involution computed by the first calling sequence for the command CartanInvolution.
We remark that, conversely, given a Cartan involution Θ, the +1, -1 eigenspaces E1 = 𝔭 and E−1= 𝔱 yield a Cartan decomposition. Also, any two Cartan involutions Θ1 and Θ2 on g are related by an inner automorphism φ : 𝔤 → 𝔤, that is, Θ2= φ Θ1φ−1 .
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 𝔤C of the Lie algebra 𝔤. This is a real semi-simple Lie algebra of twice the dimension of 𝔤 . Let σ : 𝔤C →𝔤C denote the standard conjugation map. Next use the command SplitAndCompactForms to find a complex basis of 𝔤 which defines a compact form 𝔲 of 𝔤. Identify 𝔲 with a subalgebra of 𝔤C and let τ be the corresponding conjugate map. One proves that τ is a Cartan involution of 𝔤C . If τ restricts to a mapping τ: 𝔤 →𝔤, then τ would be the required Cartan involution for 𝔤. However, this generally is not the case so the idea to conjugate τ to another Cartan involution which does restrict to 𝔤. Note that the requirement that τ restricts to a mapping τ: 𝔤 →𝔤 is equivalent to the requirement that τ commutes with σ. One proves that ψ = σ τ σ τ is a linear transformation with positive eigenvalues. The required Cartan involution is then Θ=ψ1/4τ ψ−1/4. See A.Cap and J. Slovak, Parabolic Geometries I - Background and General Theory, page 203 for further details.
We find a Cartan involution for so3,2, the Lie algebra of 5×5 matrices which are skew-symmetric with respect to the quadratic form 0I20I200001 .
LD := SimpleLieAlgebraData("so(3, 2)", so32, labelformat = "gl", labels = ['E', 'omega']):
Lie algebra: so32
The explicit matrices defining so3, 2 are
M := StandardRepresentation(so32);
From these matrices we calculate a Cartan decomposition
T, P := CartanDecomposition(M, so32);
and from this a Cartan involution Θ1
Theta1 := CartanInvolution(T, P);
We check that Θ1satisfies all the properties of a Cartan involution.
1. Θ12 = Id.
2. Θ1 is a Lie algebra homomorphism.
3. The bilinear form Bx,y= −x, Θ1y is positive-definite.
V := Tools:-DGinfo(so32, "FrameBaseVectors");
B := Matrix(10, 10, (i,j) -> Killing(-V[i], ApplyHomomorphism(Theta1, V[j])));
All of these properties are checked with the command Query/"CartanInvolution"
We calculate the Cartan involution for so3,2 using the second calling sequence. For this we need a Cartan subalgebra, the corresponding root space decomposition and a choice of positive roots.
CSA := CartanSubalgebra();
RSD := RootSpaceDecomposition(CSA);
PosRts := PositiveRoots(RSD);
Here is the Cartan involution obtained from this Cartan subalgebra.
Theta2 := CartanInvolution(CSA, RSD, PosRts);
It differs slightly from the one calculated using the first calling sequence in Example 1.
We check, by example, that if φ is an inner automorphism, then φ Θ1φ−1 is also a Cartan involution.
We use the exponential of adE35 to define φ .
A := AdjointExp(evalDG(2*E35));
phi := Transformation(so32, so32, A);
Here is the new Cartan involution.
newTheta := ComposeTransformations(phi, Theta1, InverseTransformation(phi));
Check that it works.
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