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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

 Examples

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 : gg 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 = tp , 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 E1= 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 Θ1φ1 .

• 

 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 ψ1/4. 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 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']):

DGsetup(LD);

Lie algebra: so32

(2.1)

 

The explicit matrices defining so3, 2 are

so32 > 

M := StandardRepresentation(so32);

M:=1000000000001000000000000,0100000000000000010000000,0000010000000100000000000,0000001000000000001000000,0001000100000000000000000,0000000000010001000000000,0000100000000000000000100,0000000001000000000000010,0000000000000010000010000,0000000000000000000101000

(2.2)

 

From these matrices we calculate a Cartan decomposition

so32 > 

T, P := CartanDecomposition(M, so32);

T,P:=E12E21,E14+E32,E15+E35,E25+E45,E11,E12+E21,E22,E14E32,E15E35,E25E45

(2.3)

 

and from this a Cartan involution Θ1

so32 > 

Theta1 := CartanInvolution(T, P);

Θ1:=E11,E11,E12,E21,E21,E12,E22,E22,E14,E32,E32,E14,E15,E35,E25,E45,E35,E15,E45,E25

(2.4)

 

We check that  Θ1satisfies all the properties of a Cartan involution.

1. Θ12 = Id.

so32 > 

ComposeTransformations(Theta1, Theta1);

E11,E11,E12,E12,E21,E21,E22,E22,E14,E14,E32,E32,E15,E15,E25,E25,E35,E35,E45,E45

(2.5)

 

2. Θ1  is a Lie algebra homomorphism.

so32 > 

Query(Theta1, "Homomorphism");

true

(2.6)

 

3. The bilinear form Bx,y= x, Θ1y is positive-definite.

so32 > 

V := Tools:-DGinfo(so32, "FrameBaseVectors");

V:=E11,E12,E21,E22,E14,E32,E15,E25,E35,E45

(2.7)
so32 > 

B := Matrix(10, 10, (i,j) -> Killing(-V[i], ApplyHomomorphism(Theta1, V[j])));

B:=6000000000060000000000600000000006000000000060000000000600000000006000000000060000000000600000000006

(2.8)

 

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

so32 > 

Query(Theta1, "CartanInvolution");

true

(2.9)

 

Example 2.

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();

CSA:=E11,E22

(2.10)
so32 > 

RSD := RootSpaceDecomposition(CSA);

RSD:=table1,0=E35,0,1=E45,1,1=E32,1,1=E14,1,1=E21,1,1=E12,1,0=E15,0,1=E25

(2.11)
so32 > 

PosRts := PositiveRoots(RSD);

PosRts:=11,11,10,01

(2.12)

 

Here is the Cartan involution obtained from this Cartan subalgebra.

so32 > 

Theta2 := CartanInvolution(CSA, RSD, PosRts);

Θ2:=E11,E11,E12,E21,E21,E12,E22,E22,E14,14E32,E32,4E14,E15,12E35,E25,12E45,E35,2E15,E45,2E25

(2.13)
so32 > 

Query(Theta2, "CartanInvolution");

true

(2.14)

 

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

so32 > 

Theta1;

E11,E11,E12,E21,E21,E12,E22,E22,E14,E32,E32,E14,E15,E35,E25,E45,E35,E15,E45,E25

(2.15)

 

 

 

 

Example 3.

We check, by example, that if φ is an inner automorphism, then  φ Θ1φ1 is also a Cartan involution.

 

We use the exponential of adE35 to define φ .

so32 > 

A := AdjointExp(evalDG(2*E35));

A:=1000002000010000000000102002000001000000000010000002000100020000001000000020010020000020100200000001

(2.16)
so32 > 

phi := Transformation(so32, so32, A);

φ:=E11,E11+2E35,E12,E12+2E32+2E45,E21,E21,E22,E22,E14,2E21+E142E25,E32,E32,E15,2E11+E15+2E35,E25,2E21+E25,E35,E35,E45,2E32+E45

(2.17)

 

Here is the new Cartan involution.

so32 > 

newTheta := ComposeTransformations(phi, Theta1, InverseTransformation(phi));

newTheta:=E11,5E112E156E35,E12,9E21+2E146E25,E21,E122E322E45,E22,E22,E14,2E12+9E32+6E45,E32,2E21+E142E25,E15,6E11+2E15+9E35,E25,2E12+6E32+5E45,E35,2E11+E15+2E35,E45,6E212E14+5E25

(2.18)

 

Check that it works.

so32 > 

Query(newTheta, "CartanInvolution");

true

(2.19)

See Also

Adjoint

CartanSubalgebra

CartanDecomposition

ComposeTransformations

DifferentialGeometry

InverseTransformation

Killing

LieAlgebras

Transformation

PositiveRoots

Query

RootSpaceDecomposition

StandardRepresentation

SimpleLieAlgebraData

 


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