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))
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.
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.
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):
g := CanonicalTensors("Metric", "bas",4,0);
g≔dx1⁢dx1+dx2⁢dx2+dx3⁢dx3+dx4⁢dx4
V := D_x4;
V≔D_x4
Define and initialize the general linear Lie algebra gl4.
gl4 := MatrixAlgebras("Full",4, gl4R):
DGsetup(gl4):
Calculate so4 and so3 as subalgebras of gl4.
so4_subalg := MatrixAlgebras("Subalgebra", gl4R,[g]);
so4_subalg≔e12−e21,e13−e31,e14−e41,e23−e32,e24−e42,e34−e43
so3_subalg := MatrixAlgebras("Subalgebra",gl4R,[g,V]);
so3_subalg≔e12−e21,e13−e31,e23−e32
Calculate the structure equations for so4 and find the component expressions for the vectors in so3 in terms of the vectors in so4.
g, h0 := LieAlgebraData(so4_subalg,[so3_subalg],so4);
g,h0≔e1,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
DGsetup(g);
Lie algebra: so4
Find so3 as a subalgebra of so4.
Fr := Tools:-DGinfo("FrameBaseVectors");
Fr≔e1,e2,e3,e4,e5,e6
so3 := map(DGzip,h0[1], Fr, "plus");
so3≔e1,e2,e4
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.
C := RelativeChains(so3);
C≔,,,−θ3⁢⋀⁢θ5⁢⋀⁢θ6,
Calculate the Lie algebra cohomology of so4 relative to so3.
H := Cohomology(C);
H≔,,−θ3⁢⋀⁢θ5⁢⋀⁢θ6
Show that so4 = m + so3 is a symmetric decomposition.
m := [e3,e5,e6]:
Query(so3, m, "SymmetricPair");
true
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.
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.
DGsetup([x1, x2, x3, x4, x5], E5):
g := CanonicalTensors("Metric", "bas", 5,0);
g≔dx1⁢dx1+dx2⁢dx2+dx3⁢dx3+dx4⁢dx4+dx5⁢dx5
V := D_x5;
V≔D_x5
Define and initialize the general linear Lie algebra gl5.
gl5 := MatrixAlgebras("Full", 5):
DGsetup(gl5):
Calculate so5 and so4 as subalgebras of gl5.
so5_subalg := MatrixAlgebras("Subalgebra", gl5R,[g]);
so5_subalg≔e12−e21,e13−e31,e14−e41,e15−e51,e23−e32,e24−e42,e25−e52,e34−e43,e35−e53,e45−e54
so4_subalg := MatrixAlgebras("Subalgebra",gl5R,[g,V]);
Calculate the structure equations for so5 and find the component expressions for the vectors in so4 in terms of the vectors in so5.
g, h0 := LieAlgebraData(so5_subalg,[so4_subalg],so5);
g,h0≔e1,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
Lie algebra: so5
Fr≔e1,e2,e3,e4,e5,e6,e7,e8,e9,e10
so4 := map(DGzip,h0[1], Fr, "plus");
so4≔e1,e2,e3,e5,e6,e8
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.
C := RelativeChains(so4);
C≔,,,,θ4⁢⋀⁢θ7⁢⋀⁢θ9⁢⋀⁢θ10,
Calculate the Lie algebra cohomology of so5 relative to so4.
H≔,,,θ4⁢⋀⁢θ7⁢⋀⁢θ9⁢⋀⁢θ10
Show that so5 = m + so4 is a symmetric decomposition.
M := [e4,e7,e9, e10]:
Query(so4, M, "SymmetricPair");
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.
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.
DGsetup([x1, x2, x3, y1, y2, y3], E6);
frame name: E6
g := CanonicalTensors("Metric", "bas", 6, 0);
g≔dx1⁢dx1+dx2⁢dx2+dx3⁢dx3+dy1⁢dy1+dy2⁢dy2+dy3⁢dy3
J := CanonicalTensors("ComplexStructure","bas");
J≔−dx1⁢D_y1−dx2⁢D_y2−dx3⁢D_y3+dy1⁢D_x1+dy2⁢D_x2+dy3⁢D_x3
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:
nuR := (1/2) &mult (nu &plus Tools:-DGmap(1, conjugate, nu));
nuR≔dx1⁢⋀⁢dx2⁢⋀⁢dx3−dx1⁢⋀⁢dy2⁢⋀⁢dy3+dx2⁢⋀⁢dy1⁢⋀⁢dy3−dx3⁢⋀⁢dy1⁢⋀⁢dy2
nuI := (I/2) &mult (nu &minus Tools:-DGmap(1, conjugate, nu));
nuI≔−dx1⁢⋀⁢dx2⁢⋀⁢dy3+dx1⁢⋀⁢dx3⁢⋀⁢dy2−dx2⁢⋀⁢dx3⁢⋀⁢dy1+dy1⁢⋀⁢dy2⁢⋀⁢dy3
V := D_y3;
V≔D_y3
Define and initialize the general linear Lie algebra gl6.
DGsetup(MatrixAlgebras("Full",6)):
su3_subalg := MatrixAlgebras("Subalgebra", gl6R, [g, J, nuI, nuR]);
su3_subalg≔e12−e21+e45−e54,e13−e31+e46−e64,e14−e36−e41+e63,e15+e24−e42−e51,e16+e34−e43−e61,e23−e32+e56−e65,e25−e36−e52+e63,e26+e35−e53−e62
Calculate su(3) and su(2) as subalgebras of gl6.
su2_subalg := MatrixAlgebras("Subalgebra", gl6R, [g,J,nuI,nuR,V]):
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.
Lie algebra: su3
Fr≔e1,e2,e3,e4,e5,e6,e7,e8
su2 := map(DGzip,h0[1], Fr, "plus");
su2≔e1,e3−e7,e4
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.
C := RelativeChains(su2);
C≔,θ3+θ7,−θ2⁢⋀⁢θ5−θ6⁢⋀⁢θ8,−θ2⁢⋀⁢θ6+θ5⁢⋀⁢θ8,−θ2⁢⋀⁢θ8−θ5⁢⋀⁢θ6,−θ2⁢⋀⁢θ3⁢⋀⁢θ5+θ2⁢⋀⁢θ5⁢⋀⁢θ7+θ3⁢⋀⁢θ6⁢⋀⁢θ8−θ6⁢⋀⁢θ7⁢⋀⁢θ8,−θ2⁢⋀⁢θ3⁢⋀⁢θ6+θ2⁢⋀⁢θ6⁢⋀⁢θ7−θ3⁢⋀⁢θ5⁢⋀⁢θ8+θ5⁢⋀⁢θ7⁢⋀⁢θ8,θ2⁢⋀⁢θ3⁢⋀⁢θ8+θ2⁢⋀⁢θ7⁢⋀⁢θ8−θ3⁢⋀⁢θ5⁢⋀⁢θ6−θ5⁢⋀⁢θ6⁢⋀⁢θ7,θ2⁢⋀⁢θ5⁢⋀⁢θ6⁢⋀⁢θ8,θ2⁢⋀⁢θ3⁢⋀⁢θ5⁢⋀⁢θ6⁢⋀⁢θ8+θ2⁢⋀⁢θ5⁢⋀⁢θ6⁢⋀⁢θ7⁢⋀⁢θ8,
Calculate the Lie algebra cohomology of su3 relative to su2.
H≔,,,,θ2⁢⋀⁢θ3⁢⋀⁢θ5⁢⋀⁢θ6⁢⋀⁢θ8+θ2⁢⋀⁢θ5⁢⋀⁢θ6⁢⋀⁢θ7⁢⋀⁢θ8
We calculate the general complement to su2 in su3 and use the Query program to find all possible reductive complements.
m0 := ComplementaryBasis(su2,Fr,a);
m0≔a1⁢e1+e2+a2⁢e3+a3⁢e4−a2⁢e7,a4⁢e1+a5+1⁢e3+a6⁢e4−a5⁢e7,a7⁢e1+a8⁢e3+a9⁢e4+e5−a8⁢e7,a10⁢e1+a11⁢e3+a12⁢e4+e6−a11⁢e7,a13⁢e1+a14⁢e3+a15⁢e4−a14⁢e7+e8,a1,a10,a11,a12,a13,a14,a15,a2,a3,a4,a5,a6,a7,a8,a9
TF, Eq, Soln, ReductivePairs:= Query(su2,m0,"ReductivePair");
TF,Eq,Soln,ReductivePairs≔true,0,a1,a11,a14,a15,a2,a7,a8,a9,−a10,−a11,−a12,−a13,−a14,−a2,−a3,−2⁢a4,2⁢a4,−2⁢a6,2⁢a6,−a8,−a1−2⁢a14,−a1+2⁢a9,−a10−2⁢a8,a10+2⁢a15,−a11−2⁢a3,−a11+2⁢a7,a11+2⁢a3,a11−2⁢a7,−a12+2⁢a2,a12−2⁢a13,−a13+2⁢a12,a13−2⁢a2,−a14−2⁢a1,−a14−2⁢a9,a14+2⁢a1,a14+2⁢a9,−a15−2⁢a10,−a15+2⁢a8,−a2+2⁢a12,−a2+2⁢a13,a2−2⁢a12,a2−2⁢a13,−a3−2⁢a7,a3+2⁢a11,−2⁢a5−1,2⁢a5+1,a7−2⁢a11,a7+2⁢a3,−a8−2⁢a10,−a8+2⁢a15,a8+2⁢a10,a8−2⁢a15,a9−2⁢a1,a9+2⁢a14,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,e3−e7,e4,e2,e32+e72,e5,e6,e8
Interestingly, there is a unique reductive complement but this does not make (su(3), su(2)) symmetric.
M_reductive[4];
M_reductive4
ReductivePairs[1];
e1,e3−e7,e4,e2,e32+e72,e5,e6,e8
Query(op(ReductivePairs[1]),"SymmetricPair");
false
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.
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.
DGsetup([x1, y1, u1, v1, x2, y2, u2, v2], E8):
J := CanonicalTensors("ComplexStructure", "bas");
J≔−dx1⁢D_x2−dy1⁢D_y2−du1⁢D_u2−dv1⁢D_v2+dx2⁢D_x1+dy2⁢D_y1+du2⁢D_u1+dv2⁢D_v1
K1 := evalDG( -dx1 &t D_u1 - dy1 &t D_v1 + du1 &t D_x1 + dv1 &t D_y1):
K2:= evalDG( -dx2 &t D_u2 - dy2 &t D_v2 + du2 &t D_x2 + dv2 &t D_y2):
K:= K1 &minus K2;
K≔−dx1⁢D_u1−dy1⁢D_v1+du1⁢D_x1+dv1⁢D_y1+dx2⁢D_u2+dy2⁢D_v2−du2⁢D_x2−dv2⁢D_y2
g := CanonicalTensors("Metric", "bas", 8,0);
g≔dx1⁢dx1+dy1⁢dy1+du1⁢du1+dv1⁢dv1+dx2⁢dx2+dy2⁢dy2+du2⁢du2+dv2⁢dv2
V := D_v2:
Define and initialize the general linear Lie algebra gl8.
DGsetup(MatrixAlgebras("Full", 8, gl8R)):
Calculate sp2 and sp1 as subalgebras of gl8.
sp2_subalg := MatrixAlgebras("Subalgebra", gl8R, [J, K, g]);
sp2_subalg≔e12−e21+e34−e43+e56−e65+e78−e87,e13−e31+e57−e75,e14+e23−e32−e41+e58+e67−e76−e85,e15−e37−e51+e73,e16+e25−e38−e47−e52−e61+e74+e83,e17+e35−e53−e71,e18+e27+e36+e45−e54−e63−e72−e81,e24−e42+e68−e86,e26−e48−e62+e84,e28+e46−e64−e82
sp1_subalg := MatrixAlgebras("Subalgebra", gl8R, [J, K, g, V]);
sp1_subalg≔e13−e31+e57−e75,e15−e37−e51+e73,e17+e35−e53−e71
Calculate the structure equations for sp2 and express the component expressions for the vectors in sp1 in terms of the vectors in sp2.
g, h0:= LieAlgebraData(sp2_subalg,[sp1_subalg],sp2);
g,h0≔e1,e2=−e3,e1,e3=2⁢e2−2⁢e8,e1,e4=−e5,e1,e5=2⁢e4−2⁢e9,e1,e6=−e7,e1,e7=−2⁢e10+2⁢e6,e1,e8=e3,e1,e9=e5,e1,e10=e7,e2,e3=−e1,e2,e4=−2⁢e6,e2,e5=−e7,e2,e6=2⁢e4,e2,e7=e5,e3,e4=−e7,e3,e5=−2⁢e10−2⁢e6,e3,e6=e5,e3,e7=2⁢e4+2⁢e9,e3,e8=−e1,e3,e9=−e7,e3,e10=e5,e4,e5=−e1,e4,e6=−2⁢e2,e4,e7=−e3,e5,e6=−e3,e5,e7=−2⁢e2−2⁢e8,e5,e8=e7,e5,e9=−e1,e5,e10=−e3,e6,e7=−e1,e7,e8=−e5,e7,e9=e3,e7,e10=−e1,e8,e9=−2⁢e10,e8,e10=2⁢e9,e9,e10=−2⁢e8,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0
Lie algebra: sp2
sp1 := map(DGzip,h0[1], Fr, "plus");
sp1≔e2,e4,e6
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.
C := RelativeChains(sp1);
C≔,θ8,θ9,θ10,−θ1⁢⋀⁢θ3−θ5⁢⋀⁢θ7,−θ1⁢⋀⁢θ5+θ3⁢⋀⁢θ7,−θ1⁢⋀⁢θ7−θ3⁢⋀⁢θ5,−θ8⁢⋀⁢θ9,−θ8⁢⋀⁢θ10,−θ9⁢⋀⁢θ10,−θ1⁢⋀⁢θ3⁢⋀⁢θ8−θ5⁢⋀⁢θ7⁢⋀⁢θ8,−θ1⁢⋀⁢θ5⁢⋀⁢θ8+θ3⁢⋀⁢θ7⁢⋀⁢θ8,−θ1⁢⋀⁢θ7⁢⋀⁢θ8−θ3⁢⋀⁢θ5⁢⋀⁢θ8,−θ1⁢⋀⁢θ3⁢⋀⁢θ9−θ5⁢⋀⁢θ7⁢⋀⁢θ9,−θ1⁢⋀⁢θ5⁢⋀⁢θ9+θ3⁢⋀⁢θ7⁢⋀⁢θ9,−θ1⁢⋀⁢θ7⁢⋀⁢θ9−θ3⁢⋀⁢θ5⁢⋀⁢θ9,−θ1⁢⋀⁢θ3⁢⋀⁢θ10−θ5⁢⋀⁢θ7⁢⋀⁢θ10,−θ1⁢⋀⁢θ5⁢⋀⁢θ10+θ3⁢⋀⁢θ7⁢⋀⁢θ10,−θ1⁢⋀⁢θ7⁢⋀⁢θ10−θ3⁢⋀⁢θ5⁢⋀⁢θ10,−θ8⁢⋀⁢θ9⁢⋀⁢θ10,θ1⁢⋀⁢θ7⁢⋀⁢θ9⁢⋀⁢θ10+θ3⁢⋀⁢θ5⁢⋀⁢θ9⁢⋀⁢θ10,θ1⁢⋀⁢θ3⁢⋀⁢θ5⁢⋀⁢θ7,θ1⁢⋀⁢θ3⁢⋀⁢θ8⁢⋀⁢θ9+θ5⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ9,θ1⁢⋀⁢θ5⁢⋀⁢θ8⁢⋀⁢θ9−θ3⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ9,θ1⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ9+θ3⁢⋀⁢θ5⁢⋀⁢θ8⁢⋀⁢θ9,θ1⁢⋀⁢θ3⁢⋀⁢θ8⁢⋀⁢θ10+θ5⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ10,θ1⁢⋀⁢θ5⁢⋀⁢θ8⁢⋀⁢θ10−θ3⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ10,θ1⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ10+θ3⁢⋀⁢θ5⁢⋀⁢θ8⁢⋀⁢θ10,θ1⁢⋀⁢θ3⁢⋀⁢θ9⁢⋀⁢θ10+θ5⁢⋀⁢θ7⁢⋀⁢θ9⁢⋀⁢θ10,θ1⁢⋀⁢θ5⁢⋀⁢θ9⁢⋀⁢θ10−θ3⁢⋀⁢θ7⁢⋀⁢θ9⁢⋀⁢θ10,θ1⁢⋀⁢θ3⁢⋀⁢θ5⁢⋀⁢θ7⁢⋀⁢θ8,θ1⁢⋀⁢θ3⁢⋀⁢θ5⁢⋀⁢θ7⁢⋀⁢θ9,θ1⁢⋀⁢θ3⁢⋀⁢θ5⁢⋀⁢θ7⁢⋀⁢θ10,θ1⁢⋀⁢θ3⁢⋀⁢θ8⁢⋀⁢θ9⁢⋀⁢θ10+θ5⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ9⁢⋀⁢θ10,θ1⁢⋀⁢θ5⁢⋀⁢θ8⁢⋀⁢θ9⁢⋀⁢θ10−θ3⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ9⁢⋀⁢θ10,θ1⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ9⁢⋀⁢θ10+θ3⁢⋀⁢θ5⁢⋀⁢θ8⁢⋀⁢θ9⁢⋀⁢θ10,−θ1⁢⋀⁢θ3⁢⋀⁢θ5⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ9,−θ1⁢⋀⁢θ3⁢⋀⁢θ5⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ10,−θ1⁢⋀⁢θ3⁢⋀⁢θ5⁢⋀⁢θ7⁢⋀⁢θ9⁢⋀⁢θ10,−θ1⁢⋀⁢θ3⁢⋀⁢θ5⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ9⁢⋀⁢θ10,
Calculate the Lie algebra cohomology of sp2 relative to sp1.
H≔,,,,,,−θ1⁢⋀⁢θ3⁢⋀⁢θ5⁢⋀⁢θ7⁢⋀⁢θ8⁢⋀⁢θ9⁢⋀⁢θ10
We calculate the general complement to sp1 in sp2 and use the Query program to find all possible reductive complements.
m0 := ComplementaryBasis(sp1, Fr, a);
m0≔e1+a1⁢e2+a2⁢e4+a3⁢e6,a4⁢e2+e3+a5⁢e4+a6⁢e6,a7⁢e2+a8⁢e4+e5+a9⁢e6,a10⁢e2+a11⁢e4+a12⁢e6+e7,a13⁢e2+a14⁢e4+a15⁢e6+e8,a16⁢e2+a17⁢e4+a18⁢e6+e9,a19⁢e2+a20⁢e4+a21⁢e6+e10,a1,a10,a11,a12,a13,a14,a15,a16,a17,a18,a19,a2,a20,a21,a3,a4,a5,a6,a7,a8,a9
TF, Eq, Soln, ReductivePairs:= Query(sp1, m0, "ReductivePair");
TF,Eq,Soln,ReductivePairs≔true,0,a11,a12,a4,a6,a7,a8,−a1,−a10,−2⁢a13,2⁢a13,−2⁢a14,2⁢a14,−2⁢a15,2⁢a15,−2⁢a16,2⁢a16,−2⁢a17,2⁢a17,−2⁢a18,2⁢a18,−2⁢a19,2⁢a19,−a2,−2⁢a20,2⁢a20,−2⁢a21,2⁢a21,−a3,−a5,−a9,−a1−2⁢a11,−a1+2⁢a9,a10−2⁢a2,a10+2⁢a6,−a11−2⁢a9,a11+2⁢a1,−a12+2⁢a8,a12−2⁢a4,−a2+2⁢a10,−a2−2⁢a6,−a3+2⁢a5,−a3−2⁢a7,−a4+2⁢a12,a4−2⁢a8,a5−2⁢a3,a5+2⁢a7,−a6−2⁢a10,a6+2⁢a2,−a7−2⁢a5,a7+2⁢a3,−a8+2⁢a4,a8−2⁢a12,a9−2⁢a1,a9+2⁢a11,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
e2,e4,e6,e1,e3,e5,e7,e8,e9,e10
There is a unique reductive complement but this does not make (sp2, sp1)) symmetric.
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.
Define a 7 dimensional space (on which gl(7) will act). On E7, define a 3 form phi and a vector V.
DGsetup([x1, x2, x3, x4, x5, x6, x7],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);
φ≔dx1⁢⋀⁢dx2⁢⋀⁢dx3+dx1⁢⋀⁢dx4⁢⋀⁢dx5−dx1⁢⋀⁢dx6⁢⋀⁢dx7+dx2⁢⋀⁢dx4⁢⋀⁢dx6+dx2⁢⋀⁢dx5⁢⋀⁢dx7+dx3⁢⋀⁢dx4⁢⋀⁢dx7−dx3⁢⋀⁢dx5⁢⋀⁢dx6
V:= D_x1;
V≔D_x1
Define and initialize the general linear Lie algebra gl7.
DGsetup(MatrixAlgebras("Full", 7 , gl7)):
Calculate gl7 and su3 as subalgebras of gl7.
g2_subalg := MatrixAlgebras("Subalgebra", gl7, [phi]);
g2_subalg≔e12−e21+e56−e65,e13−e31+e57−e75,e14−e36−e41+e63,e15−e37−e51+e73,e16+e34−e43−e61,e17+e35−e53−e71,e23−e32+e67−e76,e24+e35−e42−e53,e25−e34+e43−e52,e26−e37−e62+e73,e27+e36−e63−e72,e45−e54+e67−e76,e46−e57−e64+e75,e47+e56−e65−e74
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.
g, h0 := LieAlgebraData(g2_subalg,[su3_subalg],g2);
g,h0≔e1,e2=−e7,e1,e3=−e8,e1,e4=−e5−e9,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=−2⁢e6,e2,e5=e3,e2,e6=2⁢e4,e2,e7=−e1,e2,e8=e4,e2,e9=−e11−e3,e2,e10=−e6,e2,e11=e5+e9,e2,e12=−e14,e2,e14=e12,e3,e4=−e12,e3,e5=−2⁢e13−2⁢e2,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=−2⁢e2,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=e6−e8,e5,e8=−e12+e7,e5,e10=−e1+e14,e5,e11=−e13−e2,e5,e12=e6,e5,e13=−e3,e5,e14=−e4,e6,e7=−e5−e9,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=−2⁢e11,e7,e11=2⁢e10,e7,e13=−e14,e7,e14=e13,e8,e9=−2⁢e12+2⁢e7,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=−2⁢e7,e10,e12=e11,e10,e13=−e8,e10,e14=−e9,e11,e12=−e10,e11,e13=e9,e11,e14=−e8,e12,e13=−2⁢e14,e12,e14=2⁢e13,e13,e14=−2⁢e12,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1
Lie algebra: g2
Fr≔e1,e2,e3,e4,e5,e6,e7,e8,e9,e10,e11,e12,e13,e14
su3 := map(DGzip,h0[1], Fr, "plus");
su3≔e7,e8,e9,e10,e11,e12,e13,e14
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.
C := RelativeChains(su3);
C≔,,−θ1⁢⋀⁢θ2−θ3⁢⋀⁢θ4+θ5⁢⋀⁢θ6,−θ1⁢⋀⁢θ3⁢⋀⁢θ5−θ1⁢⋀⁢θ4⁢⋀⁢θ6−θ2⁢⋀⁢θ3⁢⋀⁢θ6+θ2⁢⋀⁢θ4⁢⋀⁢θ5,θ1⁢⋀⁢θ3⁢⋀⁢θ6−θ1⁢⋀⁢θ4⁢⋀⁢θ5−θ2⁢⋀⁢θ3⁢⋀⁢θ5−θ2⁢⋀⁢θ4⁢⋀⁢θ6,θ1⁢⋀⁢θ2⁢⋀⁢θ3⁢⋀⁢θ4−θ1⁢⋀⁢θ2⁢⋀⁢θ5⁢⋀⁢θ6−θ3⁢⋀⁢θ4⁢⋀⁢θ5⁢⋀⁢θ6,,−θ1⁢⋀⁢θ2⁢⋀⁢θ3⁢⋀⁢θ4⁢⋀⁢θ5⁢⋀⁢θ6,
Calculate the Lie algebra cohomology of g2 relative to su3.
H≔,,,,,−θ1⁢⋀⁢θ2⁢⋀⁢θ3⁢⋀⁢θ4⁢⋀⁢θ5⁢⋀⁢θ6
We calculate the general complement to su3 in g2 and use the Query program to find all possible reductive complements.
m0 := ComplementaryBasis(su3,Fr,a);
m0≔e1+a1⁢e7+a2⁢e8+a3⁢e9+a4⁢e10+a5⁢e11+a6⁢e12+a7⁢e13+a8⁢e14,e2+a9⁢e7+a10⁢e8+a11⁢e9+a12⁢e10+a13⁢e11+a14⁢e12+a15⁢e13+a16⁢e14,e3+a17⁢e7+a18⁢e8+a19⁢e9+a20⁢e10+a21⁢e11+a22⁢e12+a23⁢e13+a24⁢e14,e4+a25⁢e7+a26⁢e8+a27⁢e9+a28⁢e10+a29⁢e11+a30⁢e12+a31⁢e13+a32⁢e14,e5+a33⁢e7+a34⁢e8+a35⁢e9+a36⁢e10+a37⁢e11+a38⁢e12+a39⁢e13+a40⁢e14,e6+a41⁢e7+a42⁢e8+a43⁢e9+a44⁢e10+a45⁢e11+a46⁢e12+a47⁢e13+a48⁢e14,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
TF, Eq, Soln, ReductivePairs:= Query(su3,m0,"ReductivePair"):
ReductivePairs;
e7,e8,e9,e10,e11,e12,e13,e14,e1−e142,e2+e132,e3+e112,e4−e102,e5+e92,e6−e82
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.
DGsetup([x1, x2, x3, x4, x5, x6, x7], 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);
g := CanonicalTensors("Metric","bas",7,0):
DGsetup(MatrixAlgebras("Full", 7, gl7R));
Lie algebra: gl7R
so7_subalg := MatrixAlgebras("Subalgebra",gl7R,[g]):
g2_subalg := MatrixAlgebras("Subalgebra",gl7R,[phi,g]):
g, h0 := LieAlgebraData(so7_subalg, [g2_subalg],so7);
g,h0≔e1,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