Lie Algebra Cohomology - Maple Help
For the best experience, we recommend viewing online help using Google Chrome or Microsoft Edge.

Online Help

All Products    Maple    MapleSim


DifferentialGeometry Tutorials

 

Relative Lie Algebra Cohomology and the de Rham cohomology of Homogeneous Spaces.

 

 

 Overview

 Procedures Illustrated

 Example 1. The 3 sphere as SO(4)/SO(3)

 Example 2.  The 4 sphere as SO(5)/SO(4)

 Example 3.  The 5 sphere as SU(3)/SU(2)

 Example 4.  The 7 sphere as Sp(2)/Sp(1)

 Example 5.  The 6 sphere as G2/SU(3)

 Example 6.  The 7 sphere as Spin(7)/G2

 Example 7.  Complex projective space CP^4 as SU(3)/U(2)

 Example 8.  The oriented Grassmannian of 3 planes in R^6 as SO(6)/(SO(3) x SO(3))

 Overview

Let G be a compact Lie group and H a closed subgroup. Then the relative Lie algebra cohomology H^*(g, h) computes the de Rham cohomology of the homogeneous space G/H.

 

In this tutorial we shall use this result to calculate the de Rham cohomology of some classical homogeneous spaces. For each example we shall check to see if the homogeneous space under consideration is reductive or symmetric.

 Procedures Illustrated

 

In this tutorial we shall make use of the following packages and commands:

DifferentialGeometry, LieAlgebras, Tensor, DGsetup, MatrixAlgebras, CanonicalTensors, Cohomology, IntersectSubspaces, LieAlgebraData, RelativeChains 

 

We follow the method described in the help page MatrixAlgebras, the LieAlgebra Lesson on Matrix Algebras, and the Tutorials entitled Classical Matrix Algebras for constructing the Lie algebras we need in this tutorial.

 Example 1. The 3 sphere as SO(4)/SO(3)

 

In this example we construct the Lie algebra pair (g, h) = (so(4), so(3)) . The relative Lie algebra cohomology is computed and gives the cohomology of the 3 sphere. We show that (so(4), so(3)) is a symmetric pair.

 

with(DifferentialGeometry):with(LieAlgebras):with(Tensor):

 

Define a 4 dimensional space (on which gl(4) will act) and a metric tensor g and a vector V on E4.  We construct so4 as the subalgebra of gl4 which fixes g and

so3 as the subalgebra of gl4 which fixes both g and V.

DGsetup([x1, x2, x3, x4], E4):

E4 > 

g := CanonicalTensors("Metric", "bas",4,0);

gdx1dx1+dx2dx2+dx3dx3+dx4dx4

(3.1)
E4 > 

V := D_x4;

VD_x4

(3.2)

 

Define and initialize the general linear Lie algebra gl4.

E4 > 

gl4 := MatrixAlgebras("Full",4, gl4R):

E4 > 

DGsetup(gl4):

 

Calculate so4 and so3 as subalgebras of gl4.

gl4R > 

so4_subalg := MatrixAlgebras("Subalgebra", gl4R,[g]);

so4_subalge12e21,e13e31,e14e41,e23e32,e24e42,e34e43

(3.3)
gl4R > 

so3_subalg := MatrixAlgebras("Subalgebra",gl4R,[g,V]);

so3_subalge12e21,e13e31,e23e32

(3.4)

 

Calculate the structure equations for so4 and find the component expressions for the vectors in so3 in terms of the vectors in so4.

gl4R > 

g, h0 := LieAlgebraData(so4_subalg,[so3_subalg],so4);

g,h0e1,e2=e4,e1,e3=e5,e1,e4=e2,e1,e5=e3,e2,e3=e6,e2,e4=e1,e2,e6=e3,e3,e5=e1,e3,e6=e2,e4,e5=e6,e4,e6=e5,e5,e6=e4,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0

(3.5)
gl4R > 

DGsetup(g);

Lie algebra: so4

(3.6)

 

Find so3 as a subalgebra of so4.

so4 > 

Fr := Tools:-DGinfo("FrameBaseVectors");

Fre1,e2,e3,e4,e5,e6

(3.7)
so4 > 

so3 := map(DGzip,h0[1], Fr, "plus");

so3e1,e2,e4

(3.8)

 

Calculate the forms omega on so4 which satisfy Hook(X, omega) = 0 and Hook(X, d(omega)) = 0 for all X in so3.  These are the so3 relative chains.

so4 > 

C := RelativeChains(so3);

C,,,θ3θ5θ6,

(3.9)

 

Calculate the Lie algebra cohomology of so4 relative to so3.

so4 > 

H := Cohomology(C);

H,,θ3θ5θ6

(3.10)

 

Show that so4 = m + so3 is a symmetric decomposition.

so4 > 

m := [e3,e5,e6]:

so4 > 

Query(so3, m, "SymmetricPair");

true

(3.11)

 

 Example 2.  The 4 sphere as SO(5)/SO(4)

 

This example is just a higher dimensional version of Example 1.  We construct the Lie algebra pair (g, h) = (so(5), so(4)) . The relative Lie algebra cohomology is computed and gives the cohomology of the 4 sphere. We show that (so(5), so(4)) is a symmetric pair.

 

so4 > 

with(DifferentialGeometry):with(LieAlgebras):with(Tensor):

 

Define a 5 dimensional space (on which gl(5) will act) and a metric tensor g and a vector V on E5.  We construct so5 as the subalgebra of gl4 which fixes g and

so3 as the subalgebra of gl5 which fixes both g and V.

 

so4 > 

DGsetup([x1, x2, x3, x4, x5], E5):

E5 > 

g := CanonicalTensors("Metric", "bas", 5,0);

gdx1dx1+dx2dx2+dx3dx3+dx4dx4+dx5dx5

(4.1)
E5 > 

V := D_x5;

VD_x5

(4.2)

 

Define and initialize the general linear Lie algebra gl5.

E5 > 

gl5 := MatrixAlgebras("Full", 5):

E5 > 

DGsetup(gl5):

 

Calculate so5 and so4 as subalgebras of gl5.

gl5R > 

so5_subalg := MatrixAlgebras("Subalgebra", gl5R,[g]);

so5_subalge12e21,e13e31,e14e41,e15e51,e23e32,e24e42,e25e52,e34e43,e35e53,e45e54

(4.3)
gl5R > 

so4_subalg := MatrixAlgebras("Subalgebra",gl5R,[g,V]);

so4_subalge12e21,e13e31,e14e41,e23e32,e24e42,e34e43

(4.4)

 

Calculate the structure equations for so5 and find the component expressions for the vectors in so4 in terms of the vectors in so5.

gl5R > 

g, h0 := LieAlgebraData(so5_subalg,[so4_subalg],so5);

g,h0e1,e2=e5,e1,e3=e6,e1,e4=e7,e1,e5=e2,e1,e6=e3,e1,e7=e4,e2,e3=e8,e2,e4=e9,e2,e5=e1,e2,e8=e3,e2,e9=e4,e3,e4=e10,e3,e6=e1,e3,e8=e2,e3,e10=e4,e4,e7=e1,e4,e9=e2,e4,e10=e3,e5,e6=e8,e5,e7=e9,e5,e8=e6,e5,e9=e7,e6,e7=e10,e6,e8=e5,e6,e10=e7,e7,e9=e5,e7,e10=e6,e8,e9=e10,e8,e10=e9,e9,e10=e8,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0

(4.5)
gl5R > 

DGsetup(g);

Lie algebra: so5

(4.6)
so5 > 

Fr := Tools:-DGinfo("FrameBaseVectors");

Fre1,e2,e3,e4,e5,e6,e7,e8,e9,e10

(4.7)
so5 > 

so4 := map(DGzip,h0[1], Fr, "plus");

so4e1,e2,e3,e5,e6,e8

(4.8)

 

Calculate the forms omega on so5 which satisfy Hook(X, omega) = 0 and Hook(X, d(omega)) = 0 for all X in so4.  These are the so4 relative chains.

so5 > 

C := RelativeChains(so4);

C,,,,θ4θ7θ9θ10,

(4.9)

 

Calculate the Lie algebra cohomology of so5 relative to so4.

so5 > 

H := Cohomology(C);

H,,,θ4θ7θ9θ10

(4.10)

 

Show that so5 = m + so4 is a symmetric decomposition.

so5 > 

M := [e4,e7,e9, e10]:

so5 > 

Query(so4, M, "SymmetricPair");

true

(4.11)

 

 Example 3.  The 5 sphere as SU(3)/SU(2)

 

The same manifold often admits different transitive group actions leading to different realizations as homogeneous spaces. Here is a realization of the 5 sphere as the homogeneous space Su(3)/Su(2) which is different from the more familiar realization as SO(6)/SO(5).  We construct the Lie algebra pair (g, h) = (su(3), su(2)) . The relative Lie algebra cohomology is computed and gives the cohomology of the 5 sphere. We show that (su(3), su(2)) is a reductive pair but not symmetric.

 

so5 > 

with(DifferentialGeometry):with(LieAlgebras):with(Tensor):

 

Define a 6 dimensional space (on which gl(6) will act). On E6 define a metric tensor g, a complex structure J, a pair of 3 forms nuR and nuI and a vector V.  We construct su3 as the subalgebra of gl6 which fixes g, J, nuI, and nuR and su2 as the subalgebra of gl6 which also fixes V.

so5 > 

DGsetup([x1, x2, x3, y1, y2, y3], E6);

frame name: E6

(5.1)
E6 > 

g := CanonicalTensors("Metric", "bas", 6, 0);

gdx1dx1+dx2dx2+dx3dx3+dy1dy1+dy2dy2+dy3dy3

(5.2)
E6 > 

J := CanonicalTensors("ComplexStructure","bas");

Jdx1D_y1dx2D_y2dx3D_y3+dy1D_x1+dy2D_x2+dy3D_x3

(5.3)
E6 > 

dz1 := DGzip([1,I],[dx1,dy1], "plus"):
dz2 := DGzip([1,I],[dx2,dy2], "plus"):
dz3 := DGzip([1,I],[dx3,dy3], "plus"):
nu  :=dz1&wedge dz2 &wedge dz3:

E6 > 

nuR := (1/2) &mult (nu &plus Tools:-DGmap(1, conjugate, nu));

nuRdx1dx2dx3dx1dy2dy3+dx2dy1dy3dx3dy1dy2

(5.4)
E6 > 

nuI := (I/2) &mult (nu &minus Tools:-DGmap(1, conjugate, nu));

nuIdx1dx2dy3+dx1dx3dy2dx2dx3dy1+dy1dy2dy3

(5.5)
E6 > 

V := D_y3;

VD_y3

(5.6)

 

Define and initialize the general linear Lie algebra gl6.

E6 > 

DGsetup(MatrixAlgebras("Full",6)):

gl6R > 

su3_subalg := MatrixAlgebras("Subalgebra", gl6R, [g, J, nuI, nuR]);

su3_subalge12e21+e45e54,e13e31+e46e64,e14e36e41+e63,e15+e24e42e51,e16+e34e43e61,e23e32+e56e65,e25e36e52+e63,e26+e35e53e62

(5.7)

 

Calculate su(3) and su(2) as subalgebras of gl6.

gl6R > 

su2_subalg := MatrixAlgebras("Subalgebra", gl6R, [g,J,nuI,nuR,V]):

gl6R > 

g, h0 := LieAlgebraData(su3_subalg,[su2_subalg], su3):

 

Calculate the structure equations for su3 and express the component expressions for the vectors in su2 in terms of the vectors in su3.

gl6R > 

DGsetup(g);

Lie algebra: su3

(5.8)

 

su3 > 

Fr := Tools:-DGinfo("FrameBaseVectors");

Fre1,e2,e3,e4,e5,e6,e7,e8

(5.9)
su3 > 

su2 := map(DGzip,h0[1], Fr, "plus");

su2e1,e3e7,e4

(5.10)

 

Calculate the forms omega on su3 which satisfy Hook(X, omega) = 0 and Hook(X, d(omega)) = 0 for all X in su2.  These are the su2 relative chains.

su3 > 

C := RelativeChains(su2);

C,θ3+θ7,θ2θ5θ6θ8,θ2θ6+θ5θ8,θ2θ8θ5θ6,θ2θ3θ5+θ2θ5θ7+θ3θ6θ8θ6θ7θ8,θ2θ3θ6+θ2θ6θ7θ3θ5θ8+θ5θ7θ8,θ2θ3θ8+θ2θ7θ8θ3θ5θ6θ5θ6θ7,θ2θ5θ6θ8,θ2θ3θ5θ6θ8+θ2θ5θ6θ7θ8,

(5.11)

 

Calculate the Lie algebra cohomology of su3 relative to su2.

su3 > 

H := Cohomology(C);

H,,,,θ2θ3θ5θ6θ8+θ2θ5θ6θ7θ8

(5.12)

 

We calculate the general complement to su2 in su3 and use the Query program to find all possible reductive complements.

su3 > 

m0 := ComplementaryBasis(su2,Fr,a);

m0a1e1+e2+a2e3+a3e4a2e7,a4e1+a5+1e3+a6e4a5e7,a7e1+a8e3+a9e4+e5a8e7,a10e1+a11e3+a12e4+e6a11e7,a13e1+a14e3+a15e4a14e7+e8,a1,a10,a11,a12,a13,a14,a15,a2,a3,a4,a5,a6,a7,a8,a9

(5.13)
su3 > 

TF, Eq, Soln, ReductivePairs:= Query(su2,m0,"ReductivePair");

TF,Eq,Soln,ReductivePairstrue,0,a1,a11,a14,a15,a2,a7,a8,a9,a10,a11,a12,a13,a14,a2,a3,2a4,2a4,2a6,2a6,a8,a12a14,a1+2a9,a102a8,a10+2a15,a112a3,a11+2a7,a11+2a3,a112a7,a12+2a2,a122a13,a13+2a12,a132a2,a142a1,a142a9,a14+2a1,a14+2a9,a152a10,a15+2a8,a2+2a12,a2+2a13,a22a12,a22a13,a32a7,a3+2a11,2a51,2a5+1,a72a11,a7+2a3,a82a10,a8+2a15,a8+2a10,a82a15,a92a1,a9+2a14,a1=0,a10=0,a11=0,a12=0,a13=0,a14=0,a15=0,a2=0,a3=0,a4=0,a5=12,a6=0,a7=0,a8=0,a9=0,e1,e3e7,e4,e2,e32+e72,e5,e6,e8

(5.14)

 

Interestingly, there is a unique reductive complement but this does not make (su(3), su(2)) symmetric.

su3 > 

M_reductive[4];

M_reductive4

(5.15)
su3 > 

ReductivePairs[1];

e1,e3e7,e4,e2,e32+e72,e5,e6,e8

(5.16)
su3 > 

Query(op(ReductivePairs[1]),"SymmetricPair");

false

(5.17)

 Example 4.  The 7 sphere as Sp(2)/Sp(1)

 

In addition to being a SO(8) and SU(4) homogeneous space, the 7 sphere is also admits a transitive action of the symplectic group Sp(2). We construct the Lie algebra pair (g, h) = (sp(2), sp(1)) . The relative Lie algebra cohomology is computed and gives the cohomology of the 7 sphere. We show that (sp(2), su(1)) is a reductive pair but not symmetric.

 

su3 > 

with(DifferentialGeometry):with(LieAlgebras):with(Tensor):

 

Define an 8 dimensional space (on which gl(8) will act). On E8 define a metric tensor g, a pair of complex structures J and K, and a vector V.  We construct sp2 as the subalgebra of gl7 which fixes g, J, and K and sp1 as the subalgebra of gl8 which also fixes V.

 

su3 > 

DGsetup([x1, y1, u1, v1, x2, y2, u2, v2], E8):

E8 > 

J := CanonicalTensors("ComplexStructure", "bas");

Jdx1D_x2dy1D_y2du1D_u2dv1D_v2+dx2D_x1+dy2D_y1+du2D_u1+dv2D_v1

(6.1)
E8 > 

K1 := evalDG( -dx1 &t D_u1 - dy1 &t D_v1 + du1 &t D_x1 + dv1 &t D_y1):

E8 > 

K2:= evalDG( -dx2 &t D_u2 - dy2 &t D_v2 + du2 &t D_x2 + dv2 &t D_y2):

E8 > 

K:= K1 &minus K2;

Kdx1D_u1dy1D_v1+du1D_x1+dv1D_y1+dx2D_u2+dy2D_v2du2D_x2dv2D_y2

(6.2)
E8 > 

g := CanonicalTensors("Metric", "bas", 8,0);

gdx1dx1+dy1dy1+du1du1+dv1dv1+dx2dx2+dy2dy2+du2du2+dv2dv2

(6.3)
E8 > 

V := D_v2:

 

Define and initialize the general linear Lie algebra gl8.

E8 > 

DGsetup(MatrixAlgebras("Full", 8, gl8R)):

 

Calculate sp2 and sp1 as subalgebras of gl8.

gl8R > 

sp2_subalg := MatrixAlgebras("Subalgebra", gl8R, [J, K, g]);

sp2_subalge12e21+e34e43+e56e65+e78e87,e13e31+e57e75,e14+e23e32e41+e58+e67e76e85,e15e37e51+e73,e16+e25e38e47e52e61+e74+e83,e17+e35e53e71,e18+e27+e36+e45e54e63e72e81,e24e42+e68e86,e26e48e62+e84,e28+e46e64e82

(6.4)
gl8R > 

sp1_subalg := MatrixAlgebras("Subalgebra", gl8R, [J, K, g, V]);

sp1_subalge13e31+e57e75,e15e37e51+e73,e17+e35e53e71

(6.5)

 

Calculate the structure equations for sp2 and express the component expressions for the vectors in sp1 in terms of the vectors in sp2.

gl8R > 

g, h0:= LieAlgebraData(sp2_subalg,[sp1_subalg],sp2);

g,h0e1,e2=e3,e1,e3=2e22e8,e1,e4=e5,e1,e5=2e42e9,e1,e6=e7,e1,e7=2e10+2e6,e1,e8=e3,e1,e9=e5,e1,e10=e7,e2,e3=e1,e2,e4=2e6,e2,e5=e7,e2,e6=2e4,e2,e7=e5,e3,e4=e7,e3,e5=2e102e6,e3,e6=e5,e3,e7=2e4+2e9,e3,e8=e1,e3,e9=e7,e3,e10=e5,e4,e5=e1,e4,e6=2e2,e4,e7=e3,e5,e6=e3,e5,e7=2e22e8,e5,e8=e7,e5,e9=e1,e5,e10=e3,e6,e7=e1,e7,e8=e5,e7,e9=e3,e7,e10=e1,e8,e9=2e10,e8,e10=2e9,e9,e10=2e8,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0

(6.6)
gl8R > 

DGsetup(g);

Lie algebra: sp2

(6.7)
sp2 > 

Fr := Tools:-DGinfo("FrameBaseVectors");

Fre1,e2,e3,e4,e5,e6,e7,e8,e9,e10

(6.8)
sp2 > 

sp1 := map(DGzip,h0[1], Fr, "plus");

sp1e2,e4,e6

(6.9)

 

Calculate the forms omega on sp2 which satisfy Hook(X, omega) = 0 and Hook(X, d(omega)) = 0 for all X in sp1.  These are the sp1 relative chains.

sp2 > 

C := RelativeChains(sp1);

C,θ8,θ9,θ10,θ1θ3θ5θ7,θ1θ5+θ3θ7,θ1θ7θ3θ5,θ8θ9,θ8θ10,θ9θ10,θ1θ3θ8θ5θ7θ8,θ1θ5θ8+θ3θ7θ8,θ1θ7θ8θ3θ5θ8,θ1θ3θ9θ5θ7θ9,θ1θ5θ9+θ3θ7θ9,θ1θ7θ9θ3θ5θ9,θ1θ3θ10θ5θ7θ10,θ1θ5θ10+θ3θ7θ10,θ1θ7θ10θ3θ5θ10,θ8θ9θ10,θ1θ7θ9θ10+θ3θ5θ9θ10,θ1θ3θ5θ7,θ1θ3θ8θ9+θ5θ7θ8θ9,θ1θ5θ8θ9θ3θ7θ8θ9,θ1θ7θ8θ9+θ3θ5θ8θ9,θ1θ3θ8θ10+θ5θ7θ8θ10,θ1θ5θ8θ10θ3θ7θ8θ10,θ1θ7θ8θ10+θ3θ5θ8θ10,θ1θ3θ9θ10+θ5θ7θ9θ10,θ1θ5θ9θ10θ3θ7θ9θ10,θ1θ3θ5θ7θ8,θ1θ3θ5θ7θ9,θ1θ3θ5θ7θ10,θ1θ3θ8θ9θ10+θ5θ7θ8θ9θ10,θ1θ5θ8θ9θ10θ3θ7θ8θ9θ10,θ1θ7θ8θ9θ10+θ3θ5θ8θ9θ10,θ1θ3θ5θ7θ8θ9,θ1θ3θ5θ7θ8θ10,θ1θ3θ5θ7θ9θ10,θ1θ3θ5θ7θ8θ9θ10,

(6.10)

 

Calculate the Lie algebra cohomology of sp2 relative to sp1.

sp2 > 

H := Cohomology(C);

H,,,,,,θ1θ3θ5θ7θ8θ9θ10

(6.11)

 

We calculate the general complement to sp1 in sp2 and use the Query program to find all possible reductive complements.

sp2 > 

m0 := ComplementaryBasis(sp1, Fr, a);

m0e1+a1e2+a2e4+a3e6,a4e2+e3+a5e4+a6e6,a7e2+a8e4+e5+a9e6,a10e2+a11e4+a12e6+e7,a13e2+a14e4+a15e6+e8,a16e2+a17e4+a18e6+e9,a19e2+a20e4+a21e6+e10,a1,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a2,a20,a21,a3,a4,a5,a6,a7,a8,a9

(6.12)

TF, Eq, Soln, ReductivePairs:= Query(sp1, m0, "ReductivePair");

TF,Eq,Soln,ReductivePairstrue,0,a11,a12,a4,a6,a7,a8,a1,a10,2a13,2a13,2a14,2a14,2a15,2a15,2a16,2a16,2a17,2a17,2a18,2a18,2a19,2a19,a2,2a20,2a20,2a21,2a21,a3,a5,a9,a12a11,a1+2a9,a102a2,a10+2a6,a112a9,a11+2a1,a12+2a8,a122a4,a2+2a10,a22a6,a3+2a5,a32a7,a4+2a12,a42a8,a52a3,a5+2a7,a62a10,a6+2a2,a72a5,a7+2a3,a8+2a4,a82a12,a92a1,a9+2a11,a1=0,a10=0,a11=0,a12=0,a13=0,a14=0,a15=0,a16=0,a17=0,a18=0,a19=0,a2=0,a20=0,a21=0,a3=0,a4=0,a5=0,a6=0,a7=0,a8=0,a9=0,e2,e4,e6,e1,e3,e5,e7,e8,e9,e10

(6.13)
sp2 > 

ReductivePairs[1];

e2,e4,e6,e1,e3,e5,e7,e8,e9,e10

(6.14)

 

There is a unique reductive complement but this does not make (sp2, sp1)) symmetric.

sp2 > 

Query(op(ReductivePairs[1]),"SymmetricPair");

false

(6.15)

 

 Example 5.  The 6 sphere as G2/SU(3)

 

The exceptional  Lie group G2 can be defined as a subgroup of SO(7) and therefore there is a natural action of G2 on the 6 sphere. This action is transitive and the isotropy subalgebra is SU(3). In this section we compute the Lie algebra pair (g2, su3), calculate the Lie algebra cohomology of g2 relative to su3 and check that the pair (g2, su3) is reductive but not symmetric.

with(DifferentialGeometry):with(LieAlgebras):with(Tensor):

 

Define a 7 dimensional space (on which gl(7) will act). On E7, define a 3 form phi and a vector V.

sp2 > 

DGsetup([x1, x2, x3, x4, x5, x6, x7],E7):

E7 > 

phi := evalDG( dx1 &w dx2 &w dx3  + dx1 &w dx4 &w dx5 -dx1 &w dx6 &w dx7 + dx2 &w dx4 &w dx6+ dx2 &w dx5 &w dx7 +dx3 &w dx4 &w dx7 -dx3 &w dx5 &w dx6);

φdx1dx2dx3+dx1dx4dx5dx1dx6dx7+dx2dx4dx6+dx2dx5dx7+dx3dx4dx7dx3dx5dx6

(7.1)
E7 > 

V:= D_x1;

VD_x1

(7.2)

 

Define and initialize the general linear Lie algebra gl7.

E7 > 

DGsetup(MatrixAlgebras("Full", 7 , gl7)):

 

Calculate gl7 and su3 as subalgebras of gl7.

gl7 > 

g2_subalg := MatrixAlgebras("Subalgebra", gl7, [phi]);

g2_subalge12e21+e56e65,e13e31+e57e75,e14e36e41+e63,e15e37e51+e73,e16+e34e43e61,e17+e35e53e71,e23e32+e67e76,e24+e35e42e53,e25e34+e43e52,e26e37e62+e73,e27+e36e63e72,e45e54+e67e76,e46e57e64+e75,e47+e56e65e74

(7.3)
gl7 > 

su3_subalg := MatrixAlgebras("Subalgebra",gl7,[phi,V]):

 

Calculate the structure equations for g2 and find the component expressions for the vectors in su3 in terms of the vectors in g2.

gl7 > 

g, h0 :=  LieAlgebraData(g2_subalg,[su3_subalg],g2);

g,h0e1,e2=e7,e1,e3=e8,e1,e4=e5e9,e1,e5=e10+e4,e1,e6=e11,e1,e7=e2,e1,e8=e3,e1,e9=e10+e4,e1,e10=e5+e9,e1,e11=e6,e1,e12=e13,e1,e13=e12,e2,e3=e5,e2,e4=2e6,e2,e5=e3,e2,e6=2e4,e2,e7=e1,e2,e8=e4,e2,e9=e11e3,e2,e10=e6,e2,e11=e5+e9,e2,e12=e14,e2,e14=e12,e3,e4=e12,e3,e5=2e132e2,e3,e6=e14,e3,e7=e10,e3,e8=e1,e3,e9=e13+e2,e3,e10=e7,e3,e12=e4,e3,e13=e5,e3,e14=e6,e4,e5=e14,e4,e6=2e2,e4,e7=e11,e4,e8=e2,e4,e9=e1+e14,e4,e11=e7,e4,e12=e3,e4,e13=e6,e4,e14=e5,e5,e6=e12,e5,e7=e6e8,e5,e8=e12+e7,e5,e10=e1+e14,e5,e11=e13e2,e5,e12=e6,e5,e13=e3,e5,e14=e4,e6,e7=e5e9,e6,e9=e12+e7,e6,e10=e2,e6,e11=e1,e6,e12=e5,e6,e13=e4,e6,e14=e3,e7,e8=e9,e7,e9=e8,e7,e10=2e11,e7,e11=2e10,e7,e13=e14,e7,e14=e13,e8,e9=2e12+2e7,e8,e10=e13,e8,e11=e14,e8,e12=e9,e8,e13=e10,e8,e14=e11,e9,e10=e14,e9,e11=e13,e9,e12=e8,e9,e13=e11,e9,e14=e10,e10,e11=2e7,e10,e12=e11,e10,e13=e8,e10,e14=e9,e11,e12=e10,e11,e13=e9,e11,e14=e8,e12,e13=2e14,e12,e14=2e13,e13,e14=2e12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1

(7.4)
gl7 > 

DGsetup(g);

Lie algebra: g2

(7.5)
g2 > 

Fr := Tools:-DGinfo("FrameBaseVectors");

Fre1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12,e13,e14

(7.6)
g2 > 

su3 := map(DGzip,h0[1], Fr, "plus");

su3e7,e8,e9,e10,e11,e12,e13,e14

(7.7)

 

Calculate the forms omega on g2 which satisfy Hook(X, omega) = 0 and Hook(X, d(omega)) = 0 for all X in su3.  These are the su3 relative chains.

g2 > 

C := RelativeChains(su3);

C,,θ1θ2θ3θ4+θ5θ6,θ1θ3θ5θ1θ4θ6θ2θ3θ6+θ2θ4θ5,θ1θ3θ6θ1θ4θ5θ2θ3θ5θ2θ4θ6,θ1θ2θ3θ4θ1θ2θ5θ6θ3θ4θ5θ6,,θ1θ2θ3θ4θ5θ6,

(7.8)

 

Calculate the Lie algebra cohomology of g2 relative to su3.

g2 > 

H := Cohomology(C);

H,,,,,θ1θ2θ3θ4θ5θ6

(7.9)

 

We calculate the general complement to su3 in g2 and use the Query program to find all possible reductive complements.

g2 > 

m0 := ComplementaryBasis(su3,Fr,a);

m0e1+a1e7+a2e8+a3e9+a4e10+a5e11+a6e12+a7e13+a8e14,e2+a9e7+a10e8+a11e9+a12e10+a13e11+a14e12+a15e13+a16e14,e3+a17e7+a18e8+a19e9+a20e10+a21e11+a22e12+a23e13+a24e14,e4+a25e7+a26e8+a27e9+a28e10+a29e11+a30e12+a31e13+a32e14,e5+a33e7+a34e8+a35e9+a36e10+a37e11+a38e12+a39e13+a40e14,e6+a41e7+a42e8+a43e9+a44e10+a45e11+a46e12+a47e13+a48e14,a1,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a2,a20,a21,a22,a23,a24,a25,a26,a27,a28,a29,a3,a30,a31,a32,a33,a34,a35,a36,a37,a38,a39,a4,a40,a41,a42,a43,a44,a45,a46,a47,a48,a5,a6,a7,a8,a9

(7.10)
g2 > 

TF, Eq, Soln, ReductivePairs:= Query(su3,m0,"ReductivePair"):

g2 > 

ReductivePairs;

e7,e8,e9,e10,e11,e12,e13,e14,e1e142,e2+e132,e3+e112,e4e102,e5+e92,e6e82

(7.11)
g2 > 

Query(op(ReductivePairs[1]),"SymmetricPair");

false

(7.12)

 

 Example 6.  The 7 sphere as Spin(7)/G2

 

 

In this example we find that  in order to compute the relative chains , it is essential to change to a basis in so(7) adapted to gl2 to avoid serious memory problems.

 

g2 > 

with(DifferentialGeometry):with(LieAlgebras):with(Tensor):

Define and initialize the general linear Lie algebra gl7.

DGsetup([x1, x2, x3, x4, x5, x6, x7], E7):

E7 > 

phi := evalDG(dx1 &w dx2 &w dx3  + dx1 &w dx4 &w dx5 -dx1 &w dx6 &w dx7 + dx2 &w dx4 &w dx6+ dx2 &w dx5 &w dx7 +dx3 &w dx4 &w dx7 -dx3 &w dx5 &w dx6);

φdx1dx2dx3+dx1dx4dx5dx1dx6dx7+dx2dx4dx6+dx2dx5dx7+dx3dx4dx7dx3dx5dx6

(8.1)
E7 > 

g := CanonicalTensors("Metric","bas",7,0):

E7 > 

DGsetup(MatrixAlgebras("Full", 7, gl7R));

Lie algebra: gl7R

(8.2)
gl7R > 

so7_subalg := MatrixAlgebras("Subalgebra",gl7R,[g]):

gl7R > 

g2_subalg := MatrixAlgebras("Subalgebra",gl7R,[phi,g]):

gl7R > 

g, h0 := LieAlgebraData(so7_subalg, [g2_subalg],so7);

g,h0e1,e2=e7,e1,e3=e8,e1,e4=e9,e1,e5=e10,e1,e6=e11,e1,e7=e2,e1,e8=e3,e1,e9=e4,e1,e10=e5,e1,e11=e6,e2,e3=e12,e2,e4=e13,e2,e5=e14,e2,e6=e15,e2,e7=e1,e2,e12=e3,e2,e13=e4,e2,e14=e5,e2,e15=e6,e3,e4=e16,e3,e5=e17,e3,e6=e18,e3,e8=e1,e3,e12=e2,e3,e16=e4,e3,e17=e5,e3,e18=e6,e4,e5=e19,e4,e6=e20,e4,e9=e1,e4,e13=e2,e4,e16=e3,e4,e19=e5,e4,e20=e6,e5,e6=e21,e5,e10=e1,e5,e14=e2,e5,e17=e3,e5,e19=e4,e5,e21=e6,e6,e11=e1,e6,e15=e2,e6,e18=e3,e6,e20=e4,e6,e21=e5,e7,e8=e12,e7,e9=e13,e7,e10=e14,e7,e11=e15,e7,e12=e8,e7,e13=e9,e7,e14=e10,e7,e15=e11,e8,e9=e16,e8,e10=e17,e8,e11=e18,e8,e12=e7,e8,e16=e9,e8,e17=e10,e8,e18=e11,e9,e10=e19,e9,e11=e20,e9,e13=e7,e9,e16=e8,e9,e19=e10,e9,e20=e11,e10,e11=e21,e10,e14=e7,e10,e17=e8,e10,e19=e9,e10,e21=e11,e11,e15=e7,e11,e18=e8,e11,e20=e9,e11,e21=e10,e12,e13=e16,e12,e14=e17,e12,e15=e18,e12,e16=e13,e12,e17=e14,e12,e18=e15,e13,e14=e19,e13,e15=e20,e13,e16=e12,e13,e19=e14,e13,e20=e15,e14,e15=e21,e14,e17=e12,e14,e19=e13,e14,e21=e15,e15,e18=e12,e15,e20=e13,e15,e21=e14,e16,e17=e19,e16,e18=e20,e16,e19=e17,e16,e20=e18,e17,e18=e21,e17,e19=e16,e17,e21=e18,e18,e20=e16,e18,e21=e17,e19,e20=e21,e19,e21=e20,e20,e21=e19,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,0,0,0,0,0,−1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,−1,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0