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LieAlgebras[CartanInvolution] - find a Cartan involution for a non-compact, semi-simple, real Lie algebra

Calling Sequences

CartanInvolution(T$,$P)

CartanInvolution( CSA$,$RSD$,$PosRts)

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

Description

 • 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  and such that the symmetric bilinear form 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 on t and 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, 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, .
 • 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 ${\mathrm{𝔤}}_{C}$ of the Lie algebra $\mathrm{𝔤}$. This is a real semi-simple Lie algebra of twice the dimension of $\mathrm{𝔤}$ . Let  denote the standard conjugation map. Next use the command SplitAndCompactForms to find a complex basis of which defines a compact formof $\mathrm{𝔤}$. Identify with a subalgebra of ${𝔤}_{C}$ and let be the corresponding conjugate map. One proves that is a Cartan involution of ${𝔤}_{C}$ . If restricts to a mapping , then would be the required Cartan involution for $\mathrm{𝔤}$. 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 $\mathrm{σ}$. 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.

Examples

 > with(DifferentialGeometry): with(LieAlgebras):

Example 1.

We find a Cartan involution for $\mathrm{so}\left(3,2\right)$, the Lie algebra of $5×5$ matrices which are skew-symmetric with respect to the quadratic form $\left[\begin{array}{rrr}0& {I}_{2}& 0\\ {I}_{2}& 0& 0\\ 0& 0& 1\end{array}\right]$ .

 > LD := SimpleLieAlgebraData("so(3, 2)", so32, labelformat = "gl", labels = ['E', 'omega']):
 > DGsetup(LD);
 ${\mathrm{Lie algebra: so32}}$ (2.1)

The explicit matrices defining are

 so32 > M := StandardRepresentation(so32);
 ${M}{:=}\left[\left[\begin{array}{rrrrr}{1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrr}{0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrr}{0}& {0}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrr}{0}& {0}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrr}{0}& {0}& {0}& {-}{1}& {0}\\ {0}& {0}& {1}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrr}{0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {-}{1}& {0}& {0}& {0}\\ {1}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrr}{0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {-}{1}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrr}{0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {-}{1}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrr}{0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}\\ {0}& {0}& {0}& {0}& {0}\\ {-}{1}& {0}& {0}& {0}& {0}\end{array}\right]{,}\left[\begin{array}{rrrrr}{0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}\\ {0}& {-}{1}& {0}& {0}& {0}\end{array}\right]\right]$ (2.2)

From these matrices we calculate a Cartan decomposition

 so32 > T, P := CartanDecomposition(M, so32);
 ${T}{,}{P}{:=}\left[{\mathrm{E12}}{-}{\mathrm{E21}}{,}{\mathrm{E14}}{+}{\mathrm{E32}}{,}{\mathrm{E15}}{+}{\mathrm{E35}}{,}{\mathrm{E25}}{+}{\mathrm{E45}}\right]{,}\left[{\mathrm{E11}}{,}{\mathrm{E12}}{+}{\mathrm{E21}}{,}{\mathrm{E22}}{,}{\mathrm{E14}}{-}{\mathrm{E32}}{,}{\mathrm{E15}}{-}{\mathrm{E35}}{,}{\mathrm{E25}}{-}{\mathrm{E45}}\right]$ (2.3)

and from this a Cartan involution ${\mathrm{Θ}}_{1}$

 so32 > Theta1 := CartanInvolution(T, P);
 ${\mathrm{Θ1}}{:=}\left[\left[{\mathrm{E11}}{,}{-}{\mathrm{E11}}\right]{,}\left[{\mathrm{E12}}{,}{-}{\mathrm{E21}}\right]{,}\left[{\mathrm{E21}}{,}{-}{\mathrm{E12}}\right]{,}\left[{\mathrm{E22}}{,}{-}{\mathrm{E22}}\right]{,}\left[{\mathrm{E14}}{,}{\mathrm{E32}}\right]{,}\left[{\mathrm{E32}}{,}{\mathrm{E14}}\right]{,}\left[{\mathrm{E15}}{,}{\mathrm{E35}}\right]{,}\left[{\mathrm{E25}}{,}{\mathrm{E45}}\right]{,}\left[{\mathrm{E35}}{,}{\mathrm{E15}}\right]{,}\left[{\mathrm{E45}}{,}{\mathrm{E25}}\right]\right]$ (2.4)

We check that satisfies all the properties of a Cartan involution.

1. .

 so32 > ComposeTransformations(Theta1, Theta1);
 $\left[\left[{\mathrm{E11}}{,}{\mathrm{E11}}\right]{,}\left[{\mathrm{E12}}{,}{\mathrm{E12}}\right]{,}\left[{\mathrm{E21}}{,}{\mathrm{E21}}\right]{,}\left[{\mathrm{E22}}{,}{\mathrm{E22}}\right]{,}\left[{\mathrm{E14}}{,}{\mathrm{E14}}\right]{,}\left[{\mathrm{E32}}{,}{\mathrm{E32}}\right]{,}\left[{\mathrm{E15}}{,}{\mathrm{E15}}\right]{,}\left[{\mathrm{E25}}{,}{\mathrm{E25}}\right]{,}\left[{\mathrm{E35}}{,}{\mathrm{E35}}\right]{,}\left[{\mathrm{E45}}{,}{\mathrm{E45}}\right]\right]$ (2.5)

2. is a Lie algebra homomorphism.

 so32 > Query(Theta1, "Homomorphism");
 ${\mathrm{true}}$ (2.6)

3. The bilinear form is positive-definite.

 so32 > V := Tools:-DGinfo(so32, "FrameBaseVectors");
 ${V}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E12}}{,}{\mathrm{E21}}{,}{\mathrm{E22}}{,}{\mathrm{E14}}{,}{\mathrm{E32}}{,}{\mathrm{E15}}{,}{\mathrm{E25}}{,}{\mathrm{E35}}{,}{\mathrm{E45}}\right]$ (2.7)
 so32 > B := Matrix(10, 10, (i,j) -> Killing(-V[i], ApplyHomomorphism(Theta1, V[j])));
 ${B}{:=}\left[\begin{array}{rrrrrrrrrr}{6}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {6}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {6}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {6}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {6}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {6}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {6}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {6}& {0}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {6}& {0}\\ {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {6}\end{array}\right]$ (2.8)

All of these properties are checked with the the command Query/"CartanInvolution"

 so32 > Query(Theta1, "CartanInvolution");
 ${\mathrm{true}}$ (2.9)

Example 2.

We calculate the Cartan involution for $\mathrm{so}\left(3,2\right)$ 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();
 ${\mathrm{CSA}}{:=}\left[{\mathrm{E11}}{,}{\mathrm{E22}}\right]$ (2.10)
 so32 > RSD := RootSpaceDecomposition(CSA);
 ${\mathrm{RSD}}{:=}{\mathrm{table}}\left(\left[\left[{-}{1}{,}{0}\right]{=}{\mathrm{E35}}{,}\left[{0}{,}{-}{1}\right]{=}{\mathrm{E45}}{,}\left[{-}{1}{,}{-}{1}\right]{=}{\mathrm{E32}}{,}\left[{1}{,}{1}\right]{=}{\mathrm{E14}}{,}\left[{-}{1}{,}{1}\right]{=}{\mathrm{E21}}{,}\left[{1}{,}{-}{1}\right]{=}{\mathrm{E12}}{,}\left[{1}{,}{0}\right]{=}{\mathrm{E15}}{,}\left[{0}{,}{1}\right]{=}{\mathrm{E25}}\right]\right)$ (2.11)
 so32 > PosRts := PositiveRoots(RSD);
 ${\mathrm{PosRts}}{:=}\left[\left[\begin{array}{r}{1}\\ {1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {-}{1}\end{array}\right]{,}\left[\begin{array}{r}{1}\\ {0}\end{array}\right]{,}\left[\begin{array}{r}{0}\\ {1}\end{array}\right]\right]$ (2.12)

Here is the Cartan involution obtained from this Cartan subalgebra.

 so32 > Theta2 := CartanInvolution(CSA, RSD, PosRts);
 ${\mathrm{Θ2}}{:=}\left[\left[{\mathrm{E11}}{,}{-}{\mathrm{E11}}\right]{,}\left[{\mathrm{E12}}{,}{-}{\mathrm{E21}}\right]{,}\left[{\mathrm{E21}}{,}{-}{\mathrm{E12}}\right]{,}\left[{\mathrm{E22}}{,}{-}{\mathrm{E22}}\right]{,}\left[{\mathrm{E14}}{,}\frac{{1}}{{4}}{}{\mathrm{E32}}\right]{,}\left[{\mathrm{E32}}{,}{4}{}{\mathrm{E14}}\right]{,}\left[{\mathrm{E15}}{,}\frac{{1}}{{2}}{}{\mathrm{E35}}\right]{,}\left[{\mathrm{E25}}{,}\frac{{1}}{{2}}{}{\mathrm{E45}}\right]{,}\left[{\mathrm{E35}}{,}{2}{}{\mathrm{E15}}\right]{,}\left[{\mathrm{E45}}{,}{2}{}{\mathrm{E25}}\right]\right]$ (2.13)
 so32 > Query(Theta2, "CartanInvolution");
 ${\mathrm{true}}$ (2.14)

It differs slightly from the one calculated using the first calling sequence in Example 1.

 so32 > Theta1;
 $\left[\left[{\mathrm{E11}}{,}{-}{\mathrm{E11}}\right]{,}\left[{\mathrm{E12}}{,}{-}{\mathrm{E21}}\right]{,}\left[{\mathrm{E21}}{,}{-}{\mathrm{E12}}\right]{,}\left[{\mathrm{E22}}{,}{-}{\mathrm{E22}}\right]{,}\left[{\mathrm{E14}}{,}{\mathrm{E32}}\right]{,}\left[{\mathrm{E32}}{,}{\mathrm{E14}}\right]{,}\left[{\mathrm{E15}}{,}{\mathrm{E35}}\right]{,}\left[{\mathrm{E25}}{,}{\mathrm{E45}}\right]{,}\left[{\mathrm{E35}}{,}{\mathrm{E15}}\right]{,}\left[{\mathrm{E45}}{,}{\mathrm{E25}}\right]\right]$ (2.15)

Example 3.

We check, by example, that if is an inner automorphism, then is also a Cartan involution.

We use the exponential of $\mathrm{ad}\left({E}_{35}\right)$ to define

 ${A}{:=}\left[\begin{array}{rrrrrrrrrr}{1}& {0}& {0}& {0}& {0}& {0}& {2}& {0}& {0}& {0}\\ {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {1}& {0}& {-}{2}& {0}& {0}& {2}& {0}& {0}\\ {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}& {0}& {0}\\ {0}& {2}& {0}& {0}& {0}& {1}& {0}& {0}& {0}& {2}\\ {0}& {0}& {0}& {0}& {0}& {0}& {1}& {0}& {0}& {0}\\ {0}& {0}& {0}& {0}& {-}{2}& {0}& {0}& {1}& {0}& {0}\\ {2}& {0}& {0}& {0}& {0}& {0}& {2}& {0}& {1}& {0}\\ {0}& {2}& {0}& {0}& {0}& {0}& {0}& {0}& {0}& {1}\end{array}\right]$ (2.16)
 so32 > phi := Transformation(so32, so32, A);
 ${\mathrm{φ}}{:=}\left[\left[{\mathrm{E11}}{,}{\mathrm{E11}}{+}{2}{}{\mathrm{E35}}\right]{,}\left[{\mathrm{E12}}{,}{\mathrm{E12}}{+}{2}{}{\mathrm{E32}}{+}{2}{}{\mathrm{E45}}\right]{,}\left[{\mathrm{E21}}{,}{\mathrm{E21}}\right]{,}\left[{\mathrm{E22}}{,}{\mathrm{E22}}\right]{,}\left[{\mathrm{E14}}{,}{-}{2}{}{\mathrm{E21}}{+}{\mathrm{E14}}{-}{2}{}{\mathrm{E25}}\right]{,}\left[{\mathrm{E32}}{,}{\mathrm{E32}}\right]{,}\left[{\mathrm{E15}}{,}{2}{}{\mathrm{E11}}{+}{\mathrm{E15}}{+}{2}{}{\mathrm{E35}}\right]{,}\left[{\mathrm{E25}}{,}{2}{}{\mathrm{E21}}{+}{\mathrm{E25}}\right]{,}\left[{\mathrm{E35}}{,}{\mathrm{E35}}\right]{,}\left[{\mathrm{E45}}{,}{2}{}{\mathrm{E32}}{+}{\mathrm{E45}}\right]\right]$ (2.17)

Here is the new Cartan involution.

 so32 > newTheta := ComposeTransformations(phi, Theta1, InverseTransformation(phi));
 ${\mathrm{newTheta}}{:=}\left[\left[{\mathrm{E11}}{,}{-}{5}{}{\mathrm{E11}}{-}{2}{}{\mathrm{E15}}{-}{6}{}{\mathrm{E35}}\right]{,}\left[{\mathrm{E12}}{,}{-}{9}{}{\mathrm{E21}}{+}{2}{}{\mathrm{E14}}{-}{6}{}{\mathrm{E25}}\right]{,}\left[{\mathrm{E21}}{,}{-}{\mathrm{E12}}{-}{2}{}{\mathrm{E32}}{-}{2}{}{\mathrm{E45}}\right]{,}\left[{\mathrm{E22}}{,}{-}{\mathrm{E22}}\right]{,}\left[{\mathrm{E14}}{,}{2}{}{\mathrm{E12}}{+}{9}{}{\mathrm{E32}}{+}{6}{}{\mathrm{E45}}\right]{,}\left[{\mathrm{E32}}{,}{-}{2}{}{\mathrm{E21}}{+}{\mathrm{E14}}{-}{2}{}{\mathrm{E25}}\right]{,}\left[{\mathrm{E15}}{,}{6}{}{\mathrm{E11}}{+}{2}{}{\mathrm{E15}}{+}{9}{}{\mathrm{E35}}\right]{,}\left[{\mathrm{E25}}{,}{2}{}{\mathrm{E12}}{+}{6}{}{\mathrm{E32}}{+}{5}{}{\mathrm{E45}}\right]{,}\left[{\mathrm{E35}}{,}{2}{}{\mathrm{E11}}{+}{\mathrm{E15}}{+}{2}{}{\mathrm{E35}}\right]{,}\left[{\mathrm{E45}}{,}{6}{}{\mathrm{E21}}{-}{2}{}{\mathrm{E14}}{+}{5}{}{\mathrm{E25}}\right]\right]$ (2.18)

Check that it works.

 so32 > Query(newTheta, "CartanInvolution");
 ${\mathrm{true}}$ (2.19)