Lie Algebra Cohomology - Maple Help

DifferentialGeometry Tutorials

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

 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:   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);
 ${g}{≔}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\mathrm{dx3}}{+}{\mathrm{dx4}}{}{\mathrm{dx4}}$ (3.1)
 E4 > V := D_x4;
 ${V}{≔}{\mathrm{D_x4}}$ (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]);
 ${\mathrm{so4_subalg}}{≔}\left[{\mathrm{e12}}{-}{\mathrm{e21}}{,}{\mathrm{e13}}{-}{\mathrm{e31}}{,}{\mathrm{e14}}{-}{\mathrm{e41}}{,}{\mathrm{e23}}{-}{\mathrm{e32}}{,}{\mathrm{e24}}{-}{\mathrm{e42}}{,}{\mathrm{e34}}{-}{\mathrm{e43}}\right]$ (3.3)
 gl4R > so3_subalg := MatrixAlgebras("Subalgebra",gl4R,[g,V]);
 ${\mathrm{so3_subalg}}{≔}\left[{\mathrm{e12}}{-}{\mathrm{e21}}{,}{\mathrm{e13}}{-}{\mathrm{e31}}{,}{\mathrm{e23}}{-}{\mathrm{e32}}\right]$ (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);
 ${g}{,}{\mathrm{h0}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e4}}\right]{,}\left[\left[\left[{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}\right]\right]\right]$ (3.5)
 gl4R > DGsetup(g);
 ${\mathrm{Lie algebra: so4}}$ (3.6)

Find so3 as a subalgebra of so4.

 so4 > Fr := Tools:-DGinfo("FrameBaseVectors");
 ${\mathrm{Fr}}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}\right]$ (3.7)
 so4 > so3 := map(DGzip,h0[1], Fr, "plus");
 ${\mathrm{so3}}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e4}}\right]$ (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);
 ${C}{≔}\left[\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}\right]{,}\left[\right]\right]$ (3.9)

Calculate the Lie algebra cohomology of so4 relative to so3.

 so4 > H := Cohomology(C);
 ${H}{≔}\left[\left[\right]{,}\left[\right]{,}\left[{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}\right]\right]$ (3.10)

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

 so4 > m := [e3,e5,e6]:
 so4 > Query(so3, m, "SymmetricPair");
 ${\mathrm{true}}$ (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);
 ${g}{≔}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\mathrm{dx3}}{+}{\mathrm{dx4}}{}{\mathrm{dx4}}{+}{\mathrm{dx5}}{}{\mathrm{dx5}}$ (4.1)
 E5 > V := D_x5;
 ${V}{≔}{\mathrm{D_x5}}$ (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]);
 ${\mathrm{so5_subalg}}{≔}\left[{\mathrm{e12}}{-}{\mathrm{e21}}{,}{\mathrm{e13}}{-}{\mathrm{e31}}{,}{\mathrm{e14}}{-}{\mathrm{e41}}{,}{\mathrm{e15}}{-}{\mathrm{e51}}{,}{\mathrm{e23}}{-}{\mathrm{e32}}{,}{\mathrm{e24}}{-}{\mathrm{e42}}{,}{\mathrm{e25}}{-}{\mathrm{e52}}{,}{\mathrm{e34}}{-}{\mathrm{e43}}{,}{\mathrm{e35}}{-}{\mathrm{e53}}{,}{\mathrm{e45}}{-}{\mathrm{e54}}\right]$ (4.3)
 gl5R > so4_subalg := MatrixAlgebras("Subalgebra",gl5R,[g,V]);
 ${\mathrm{so4_subalg}}{≔}\left[{\mathrm{e12}}{-}{\mathrm{e21}}{,}{\mathrm{e13}}{-}{\mathrm{e31}}{,}{\mathrm{e14}}{-}{\mathrm{e41}}{,}{\mathrm{e23}}{-}{\mathrm{e32}}{,}{\mathrm{e24}}{-}{\mathrm{e42}}{,}{\mathrm{e34}}{-}{\mathrm{e43}}\right]$ (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}{,}{\mathrm{h0}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e8}}\right]{,}\left[\left[\left[{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}\right]\right]\right]$ (4.5)
 gl5R > DGsetup(g);
 ${\mathrm{Lie algebra: so5}}$ (4.6)
 so5 > Fr := Tools:-DGinfo("FrameBaseVectors");
 ${\mathrm{Fr}}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}{,}{\mathrm{e8}}{,}{\mathrm{e9}}{,}{\mathrm{e10}}\right]$ (4.7)
 so5 > so4 := map(DGzip,h0[1], Fr, "plus");
 ${\mathrm{so4}}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e8}}\right]$ (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);
 ${C}{≔}\left[\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}\right]{,}\left[\right]\right]$ (4.9)

Calculate the Lie algebra cohomology of so5 relative to so4.

 so5 > H := Cohomology(C);
 ${H}{≔}\left[\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}\right]\right]$ (4.10)

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

 so5 > M := [e4,e7,e9, e10]:
 so5 > Query(so4, M, "SymmetricPair");
 ${\mathrm{true}}$ (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);
 ${\mathrm{frame name: E6}}$ (5.1)
 E6 > g := CanonicalTensors("Metric", "bas", 6, 0);
 ${g}{≔}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dx3}}{}{\mathrm{dx3}}{+}{\mathrm{dy1}}{}{\mathrm{dy1}}{+}{\mathrm{dy2}}{}{\mathrm{dy2}}{+}{\mathrm{dy3}}{}{\mathrm{dy3}}$ (5.2)
 E6 > J := CanonicalTensors("ComplexStructure","bas");
 ${J}{≔}{-}{\mathrm{dx1}}{}{\mathrm{D_y1}}{-}{\mathrm{dx2}}{}{\mathrm{D_y2}}{-}{\mathrm{dx3}}{}{\mathrm{D_y3}}{+}{\mathrm{dy1}}{}{\mathrm{D_x1}}{+}{\mathrm{dy2}}{}{\mathrm{D_x2}}{+}{\mathrm{dy3}}{}{\mathrm{D_x3}}$ (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));
 ${\mathrm{nuR}}{≔}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx3}}{-}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dy2}}{}{\bigwedge }{}{\mathrm{dy3}}{+}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dy1}}{}{\bigwedge }{}{\mathrm{dy3}}{-}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dy1}}{}{\bigwedge }{}{\mathrm{dy2}}$ (5.4)
 E6 > nuI := (I/2) &mult (nu &minus Tools:-DGmap(1, conjugate, nu));
 ${\mathrm{nuI}}{≔}{-}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dy3}}{+}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dy2}}{-}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dy1}}{+}{\mathrm{dy1}}{}{\bigwedge }{}{\mathrm{dy2}}{}{\bigwedge }{}{\mathrm{dy3}}$ (5.5)
 E6 > V := D_y3;
 ${V}{≔}{\mathrm{D_y3}}$ (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]);
 ${\mathrm{su3_subalg}}{≔}\left[{\mathrm{e12}}{-}{\mathrm{e21}}{+}{\mathrm{e45}}{-}{\mathrm{e54}}{,}{\mathrm{e13}}{-}{\mathrm{e31}}{+}{\mathrm{e46}}{-}{\mathrm{e64}}{,}{\mathrm{e14}}{-}{\mathrm{e36}}{-}{\mathrm{e41}}{+}{\mathrm{e63}}{,}{\mathrm{e15}}{+}{\mathrm{e24}}{-}{\mathrm{e42}}{-}{\mathrm{e51}}{,}{\mathrm{e16}}{+}{\mathrm{e34}}{-}{\mathrm{e43}}{-}{\mathrm{e61}}{,}{\mathrm{e23}}{-}{\mathrm{e32}}{+}{\mathrm{e56}}{-}{\mathrm{e65}}{,}{\mathrm{e25}}{-}{\mathrm{e36}}{-}{\mathrm{e52}}{+}{\mathrm{e63}}{,}{\mathrm{e26}}{+}{\mathrm{e35}}{-}{\mathrm{e53}}{-}{\mathrm{e62}}\right]$ (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);
 ${\mathrm{Lie algebra: su3}}$ (5.8)

 su3 > Fr := Tools:-DGinfo("FrameBaseVectors");
 ${\mathrm{Fr}}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}{,}{\mathrm{e8}}\right]$ (5.9)
 su3 > su2 := map(DGzip,h0[1], Fr, "plus");
 ${\mathrm{su2}}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e3}}{-}{\mathrm{e7}}{,}{\mathrm{e4}}\right]$ (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);
 ${C}{≔}\left[\left[\right]{,}\left[{\mathrm{θ3}}{+}{\mathrm{θ7}}\right]{,}\left[{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ5}}{-}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ8}}{,}{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ6}}{+}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ8}}{,}{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ8}}{-}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}\right]{,}\left[{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{+}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{+}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ8}}{-}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{,}{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ6}}{+}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ7}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ8}}{+}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{,}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ8}}{+}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}{-}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ7}}\right]{,}\left[{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ8}}\right]{,}\left[{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ8}}{+}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}\right]{,}\left[\right]\right]$ (5.11)

Calculate the Lie algebra cohomology of su3 relative to su2.

 su3 > H := Cohomology(C);
 ${H}{≔}\left[\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ8}}{+}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}\right]\right]$ (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);
 ${\mathrm{m0}}{≔}\left[{\mathrm{a1}}{}{\mathrm{e1}}{+}{\mathrm{e2}}{+}{\mathrm{a2}}{}{\mathrm{e3}}{+}{\mathrm{a3}}{}{\mathrm{e4}}{-}{\mathrm{a2}}{}{\mathrm{e7}}{,}{\mathrm{a4}}{}{\mathrm{e1}}{+}\left({\mathrm{a5}}{+}{1}\right){}{\mathrm{e3}}{+}{\mathrm{a6}}{}{\mathrm{e4}}{-}{\mathrm{a5}}{}{\mathrm{e7}}{,}{\mathrm{a7}}{}{\mathrm{e1}}{+}{\mathrm{a8}}{}{\mathrm{e3}}{+}{\mathrm{a9}}{}{\mathrm{e4}}{+}{\mathrm{e5}}{-}{\mathrm{a8}}{}{\mathrm{e7}}{,}{\mathrm{a10}}{}{\mathrm{e1}}{+}{\mathrm{a11}}{}{\mathrm{e3}}{+}{\mathrm{a12}}{}{\mathrm{e4}}{+}{\mathrm{e6}}{-}{\mathrm{a11}}{}{\mathrm{e7}}{,}{\mathrm{a13}}{}{\mathrm{e1}}{+}{\mathrm{a14}}{}{\mathrm{e3}}{+}{\mathrm{a15}}{}{\mathrm{e4}}{-}{\mathrm{a14}}{}{\mathrm{e7}}{+}{\mathrm{e8}}\right]{,}\left\{{\mathrm{a1}}{,}{\mathrm{a10}}{,}{\mathrm{a11}}{,}{\mathrm{a12}}{,}{\mathrm{a13}}{,}{\mathrm{a14}}{,}{\mathrm{a15}}{,}{\mathrm{a2}}{,}{\mathrm{a3}}{,}{\mathrm{a4}}{,}{\mathrm{a5}}{,}{\mathrm{a6}}{,}{\mathrm{a7}}{,}{\mathrm{a8}}{,}{\mathrm{a9}}\right\}$ (5.13)
 su3 > TF, Eq, Soln, ReductivePairs:= Query(su2,m0,"ReductivePair");
 ${\mathrm{TF}}{,}{\mathrm{Eq}}{,}{\mathrm{Soln}}{,}{\mathrm{ReductivePairs}}{≔}{\mathrm{true}}{,}\left\{{0}{,}{\mathrm{a1}}{,}{\mathrm{a11}}{,}{\mathrm{a14}}{,}{\mathrm{a15}}{,}{\mathrm{a2}}{,}{\mathrm{a7}}{,}{\mathrm{a8}}{,}{\mathrm{a9}}{,}{-}{\mathrm{a10}}{,}{-}{\mathrm{a11}}{,}{-}{\mathrm{a12}}{,}{-}{\mathrm{a13}}{,}{-}{\mathrm{a14}}{,}{-}{\mathrm{a2}}{,}{-}{\mathrm{a3}}{,}{-}{2}{}{\mathrm{a4}}{,}{2}{}{\mathrm{a4}}{,}{-}{2}{}{\mathrm{a6}}{,}{2}{}{\mathrm{a6}}{,}{-}{\mathrm{a8}}{,}{-}{\mathrm{a1}}{-}{2}{}{\mathrm{a14}}{,}{-}{\mathrm{a1}}{+}{2}{}{\mathrm{a9}}{,}{-}{\mathrm{a10}}{-}{2}{}{\mathrm{a8}}{,}{\mathrm{a10}}{+}{2}{}{\mathrm{a15}}{,}{-}{\mathrm{a11}}{-}{2}{}{\mathrm{a3}}{,}{-}{\mathrm{a11}}{+}{2}{}{\mathrm{a7}}{,}{\mathrm{a11}}{+}{2}{}{\mathrm{a3}}{,}{\mathrm{a11}}{-}{2}{}{\mathrm{a7}}{,}{-}{\mathrm{a12}}{+}{2}{}{\mathrm{a2}}{,}{\mathrm{a12}}{-}{2}{}{\mathrm{a13}}{,}{-}{\mathrm{a13}}{+}{2}{}{\mathrm{a12}}{,}{\mathrm{a13}}{-}{2}{}{\mathrm{a2}}{,}{-}{\mathrm{a14}}{-}{2}{}{\mathrm{a1}}{,}{-}{\mathrm{a14}}{-}{2}{}{\mathrm{a9}}{,}{\mathrm{a14}}{+}{2}{}{\mathrm{a1}}{,}{\mathrm{a14}}{+}{2}{}{\mathrm{a9}}{,}{-}{\mathrm{a15}}{-}{2}{}{\mathrm{a10}}{,}{-}{\mathrm{a15}}{+}{2}{}{\mathrm{a8}}{,}{-}{\mathrm{a2}}{+}{2}{}{\mathrm{a12}}{,}{-}{\mathrm{a2}}{+}{2}{}{\mathrm{a13}}{,}{\mathrm{a2}}{-}{2}{}{\mathrm{a12}}{,}{\mathrm{a2}}{-}{2}{}{\mathrm{a13}}{,}{-}{\mathrm{a3}}{-}{2}{}{\mathrm{a7}}{,}{\mathrm{a3}}{+}{2}{}{\mathrm{a11}}{,}{-}{2}{}{\mathrm{a5}}{-}{1}{,}{2}{}{\mathrm{a5}}{+}{1}{,}{\mathrm{a7}}{-}{2}{}{\mathrm{a11}}{,}{\mathrm{a7}}{+}{2}{}{\mathrm{a3}}{,}{-}{\mathrm{a8}}{-}{2}{}{\mathrm{a10}}{,}{-}{\mathrm{a8}}{+}{2}{}{\mathrm{a15}}{,}{\mathrm{a8}}{+}{2}{}{\mathrm{a10}}{,}{\mathrm{a8}}{-}{2}{}{\mathrm{a15}}{,}{\mathrm{a9}}{-}{2}{}{\mathrm{a1}}{,}{\mathrm{a9}}{+}{2}{}{\mathrm{a14}}\right\}{,}\left[\left\{{\mathrm{a1}}{=}{0}{,}{\mathrm{a10}}{=}{0}{,}{\mathrm{a11}}{=}{0}{,}{\mathrm{a12}}{=}{0}{,}{\mathrm{a13}}{=}{0}{,}{\mathrm{a14}}{=}{0}{,}{\mathrm{a15}}{=}{0}{,}{\mathrm{a2}}{=}{0}{,}{\mathrm{a3}}{=}{0}{,}{\mathrm{a4}}{=}{0}{,}{\mathrm{a5}}{=}{-}\frac{{1}}{{2}}{,}{\mathrm{a6}}{=}{0}{,}{\mathrm{a7}}{=}{0}{,}{\mathrm{a8}}{=}{0}{,}{\mathrm{a9}}{=}{0}\right\}\right]{,}\left[\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}{-}{\mathrm{e7}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e2}}{,}\frac{{\mathrm{e3}}}{{2}}{+}\frac{{\mathrm{e7}}}{{2}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e8}}\right]\right]\right]$ (5.14)

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

 su3 > M_reductive[4];
 ${{\mathrm{M_reductive}}}_{{4}}$ (5.15)
 su3 > ReductivePairs[1];
 $\left[\left[{\mathrm{e1}}{,}{\mathrm{e3}}{-}{\mathrm{e7}}{,}{\mathrm{e4}}\right]{,}\left[{\mathrm{e2}}{,}\frac{{\mathrm{e3}}}{{2}}{+}\frac{{\mathrm{e7}}}{{2}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e8}}\right]\right]$ (5.16)
 su3 > Query(op(ReductivePairs[1]),"SymmetricPair");
 ${\mathrm{false}}$ (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");
 ${J}{≔}{-}{\mathrm{dx1}}{}{\mathrm{D_x2}}{-}{\mathrm{dy1}}{}{\mathrm{D_y2}}{-}{\mathrm{du1}}{}{\mathrm{D_u2}}{-}{\mathrm{dv1}}{}{\mathrm{D_v2}}{+}{\mathrm{dx2}}{}{\mathrm{D_x1}}{+}{\mathrm{dy2}}{}{\mathrm{D_y1}}{+}{\mathrm{du2}}{}{\mathrm{D_u1}}{+}{\mathrm{dv2}}{}{\mathrm{D_v1}}$ (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;
 ${K}{≔}{-}{\mathrm{dx1}}{}{\mathrm{D_u1}}{-}{\mathrm{dy1}}{}{\mathrm{D_v1}}{+}{\mathrm{du1}}{}{\mathrm{D_x1}}{+}{\mathrm{dv1}}{}{\mathrm{D_y1}}{+}{\mathrm{dx2}}{}{\mathrm{D_u2}}{+}{\mathrm{dy2}}{}{\mathrm{D_v2}}{-}{\mathrm{du2}}{}{\mathrm{D_x2}}{-}{\mathrm{dv2}}{}{\mathrm{D_y2}}$ (6.2)
 E8 > g := CanonicalTensors("Metric", "bas", 8,0);
 ${g}{≔}{\mathrm{dx1}}{}{\mathrm{dx1}}{+}{\mathrm{dy1}}{}{\mathrm{dy1}}{+}{\mathrm{du1}}{}{\mathrm{du1}}{+}{\mathrm{dv1}}{}{\mathrm{dv1}}{+}{\mathrm{dx2}}{}{\mathrm{dx2}}{+}{\mathrm{dy2}}{}{\mathrm{dy2}}{+}{\mathrm{du2}}{}{\mathrm{du2}}{+}{\mathrm{dv2}}{}{\mathrm{dv2}}$ (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]);
 ${\mathrm{sp2_subalg}}{≔}\left[{\mathrm{e12}}{-}{\mathrm{e21}}{+}{\mathrm{e34}}{-}{\mathrm{e43}}{+}{\mathrm{e56}}{-}{\mathrm{e65}}{+}{\mathrm{e78}}{-}{\mathrm{e87}}{,}{\mathrm{e13}}{-}{\mathrm{e31}}{+}{\mathrm{e57}}{-}{\mathrm{e75}}{,}{\mathrm{e14}}{+}{\mathrm{e23}}{-}{\mathrm{e32}}{-}{\mathrm{e41}}{+}{\mathrm{e58}}{+}{\mathrm{e67}}{-}{\mathrm{e76}}{-}{\mathrm{e85}}{,}{\mathrm{e15}}{-}{\mathrm{e37}}{-}{\mathrm{e51}}{+}{\mathrm{e73}}{,}{\mathrm{e16}}{+}{\mathrm{e25}}{-}{\mathrm{e38}}{-}{\mathrm{e47}}{-}{\mathrm{e52}}{-}{\mathrm{e61}}{+}{\mathrm{e74}}{+}{\mathrm{e83}}{,}{\mathrm{e17}}{+}{\mathrm{e35}}{-}{\mathrm{e53}}{-}{\mathrm{e71}}{,}{\mathrm{e18}}{+}{\mathrm{e27}}{+}{\mathrm{e36}}{+}{\mathrm{e45}}{-}{\mathrm{e54}}{-}{\mathrm{e63}}{-}{\mathrm{e72}}{-}{\mathrm{e81}}{,}{\mathrm{e24}}{-}{\mathrm{e42}}{+}{\mathrm{e68}}{-}{\mathrm{e86}}{,}{\mathrm{e26}}{-}{\mathrm{e48}}{-}{\mathrm{e62}}{+}{\mathrm{e84}}{,}{\mathrm{e28}}{+}{\mathrm{e46}}{-}{\mathrm{e64}}{-}{\mathrm{e82}}\right]$ (6.4)
 gl8R > sp1_subalg := MatrixAlgebras("Subalgebra", gl8R, [J, K, g, V]);
 ${\mathrm{sp1_subalg}}{≔}\left[{\mathrm{e13}}{-}{\mathrm{e31}}{+}{\mathrm{e57}}{-}{\mathrm{e75}}{,}{\mathrm{e15}}{-}{\mathrm{e37}}{-}{\mathrm{e51}}{+}{\mathrm{e73}}{,}{\mathrm{e17}}{+}{\mathrm{e35}}{-}{\mathrm{e53}}{-}{\mathrm{e71}}\right]$ (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.

 ${g}{,}{\mathrm{h0}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{2}{}{\mathrm{e2}}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{2}{}{\mathrm{e4}}{-}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e10}}{+}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e10}}{-}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{2}{}{\mathrm{e4}}{+}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{-}{2}{}{\mathrm{e2}}{-}{2}{}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{-}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{2}{}{\mathrm{e9}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e8}}\right]{,}\left[\left[\left[{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}\right]\right]\right]$ (6.6)
 gl8R > DGsetup(g);
 ${\mathrm{Lie algebra: sp2}}$ (6.7)
 sp2 > Fr := Tools:-DGinfo("FrameBaseVectors");
 ${\mathrm{Fr}}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}{,}{\mathrm{e8}}{,}{\mathrm{e9}}{,}{\mathrm{e10}}\right]$ (6.8)
 sp2 > sp1 := map(DGzip,h0[1], Fr, "plus");
 ${\mathrm{sp1}}{≔}\left[{\mathrm{e2}}{,}{\mathrm{e4}}{,}{\mathrm{e6}}\right]$ (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}{≔}\left[\left[\right]{,}\left[{\mathrm{θ8}}{,}{\mathrm{θ9}}{,}{\mathrm{θ10}}\right]{,}\left[{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{-}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ5}}{+}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{,}{-}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{,}{-}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{-}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}\right]{,}\left[{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ8}}{-}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ8}}{+}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ8}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ9}}{-}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ9}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ9}}{+}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ9}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ9}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ9}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ10}}{-}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ10}}{+}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ10}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{-}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}\right]{,}\left[{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}{+}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{+}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{+}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ10}}{+}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ10}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ10}}{+}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}{+}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}\right]{,}\left[{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ9}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}{+}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}{+}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}\right]{,}\left[{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ10}}{,}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}\right]{,}\left[{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}\right]{,}\left[\right]\right]$ (6.10)

Calculate the Lie algebra cohomology of sp2 relative to sp1.

 sp2 > H := Cohomology(C);
 ${H}{≔}\left[\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ7}}{}{\bigwedge }{}{\mathrm{θ8}}{}{\bigwedge }{}{\mathrm{θ9}}{}{\bigwedge }{}{\mathrm{θ10}}\right]\right]$ (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);
 ${\mathrm{m0}}{≔}\left[{\mathrm{e1}}{+}{\mathrm{a1}}{}{\mathrm{e2}}{+}{\mathrm{a2}}{}{\mathrm{e4}}{+}{\mathrm{a3}}{}{\mathrm{e6}}{,}{\mathrm{a4}}{}{\mathrm{e2}}{+}{\mathrm{e3}}{+}{\mathrm{a5}}{}{\mathrm{e4}}{+}{\mathrm{a6}}{}{\mathrm{e6}}{,}{\mathrm{a7}}{}{\mathrm{e2}}{+}{\mathrm{a8}}{}{\mathrm{e4}}{+}{\mathrm{e5}}{+}{\mathrm{a9}}{}{\mathrm{e6}}{,}{\mathrm{a10}}{}{\mathrm{e2}}{+}{\mathrm{a11}}{}{\mathrm{e4}}{+}{\mathrm{a12}}{}{\mathrm{e6}}{+}{\mathrm{e7}}{,}{\mathrm{a13}}{}{\mathrm{e2}}{+}{\mathrm{a14}}{}{\mathrm{e4}}{+}{\mathrm{a15}}{}{\mathrm{e6}}{+}{\mathrm{e8}}{,}{\mathrm{a16}}{}{\mathrm{e2}}{+}{\mathrm{a17}}{}{\mathrm{e4}}{+}{\mathrm{a18}}{}{\mathrm{e6}}{+}{\mathrm{e9}}{,}{\mathrm{a19}}{}{\mathrm{e2}}{+}{\mathrm{a20}}{}{\mathrm{e4}}{+}{\mathrm{a21}}{}{\mathrm{e6}}{+}{\mathrm{e10}}\right]{,}\left\{{\mathrm{a1}}{,}{\mathrm{a10}}{,}{\mathrm{a11}}{,}{\mathrm{a12}}{,}{\mathrm{a13}}{,}{\mathrm{a14}}{,}{\mathrm{a15}}{,}{\mathrm{a16}}{,}{\mathrm{a17}}{,}{\mathrm{a18}}{,}{\mathrm{a19}}{,}{\mathrm{a2}}{,}{\mathrm{a20}}{,}{\mathrm{a21}}{,}{\mathrm{a3}}{,}{\mathrm{a4}}{,}{\mathrm{a5}}{,}{\mathrm{a6}}{,}{\mathrm{a7}}{,}{\mathrm{a8}}{,}{\mathrm{a9}}\right\}$ (6.12)
 > TF, Eq, Soln, ReductivePairs:= Query(sp1, m0, "ReductivePair");
 ${\mathrm{TF}}{,}{\mathrm{Eq}}{,}{\mathrm{Soln}}{,}{\mathrm{ReductivePairs}}{≔}{\mathrm{true}}{,}\left\{{0}{,}{\mathrm{a11}}{,}{\mathrm{a12}}{,}{\mathrm{a4}}{,}{\mathrm{a6}}{,}{\mathrm{a7}}{,}{\mathrm{a8}}{,}{-}{\mathrm{a1}}{,}{-}{\mathrm{a10}}{,}{-}{2}{}{\mathrm{a13}}{,}{2}{}{\mathrm{a13}}{,}{-}{2}{}{\mathrm{a14}}{,}{2}{}{\mathrm{a14}}{,}{-}{2}{}{\mathrm{a15}}{,}{2}{}{\mathrm{a15}}{,}{-}{2}{}{\mathrm{a16}}{,}{2}{}{\mathrm{a16}}{,}{-}{2}{}{\mathrm{a17}}{,}{2}{}{\mathrm{a17}}{,}{-}{2}{}{\mathrm{a18}}{,}{2}{}{\mathrm{a18}}{,}{-}{2}{}{\mathrm{a19}}{,}{2}{}{\mathrm{a19}}{,}{-}{\mathrm{a2}}{,}{-}{2}{}{\mathrm{a20}}{,}{2}{}{\mathrm{a20}}{,}{-}{2}{}{\mathrm{a21}}{,}{2}{}{\mathrm{a21}}{,}{-}{\mathrm{a3}}{,}{-}{\mathrm{a5}}{,}{-}{\mathrm{a9}}{,}{-}{\mathrm{a1}}{-}{2}{}{\mathrm{a11}}{,}{-}{\mathrm{a1}}{+}{2}{}{\mathrm{a9}}{,}{\mathrm{a10}}{-}{2}{}{\mathrm{a2}}{,}{\mathrm{a10}}{+}{2}{}{\mathrm{a6}}{,}{-}{\mathrm{a11}}{-}{2}{}{\mathrm{a9}}{,}{\mathrm{a11}}{+}{2}{}{\mathrm{a1}}{,}{-}{\mathrm{a12}}{+}{2}{}{\mathrm{a8}}{,}{\mathrm{a12}}{-}{2}{}{\mathrm{a4}}{,}{-}{\mathrm{a2}}{+}{2}{}{\mathrm{a10}}{,}{-}{\mathrm{a2}}{-}{2}{}{\mathrm{a6}}{,}{-}{\mathrm{a3}}{+}{2}{}{\mathrm{a5}}{,}{-}{\mathrm{a3}}{-}{2}{}{\mathrm{a7}}{,}{-}{\mathrm{a4}}{+}{2}{}{\mathrm{a12}}{,}{\mathrm{a4}}{-}{2}{}{\mathrm{a8}}{,}{\mathrm{a5}}{-}{2}{}{\mathrm{a3}}{,}{\mathrm{a5}}{+}{2}{}{\mathrm{a7}}{,}{-}{\mathrm{a6}}{-}{2}{}{\mathrm{a10}}{,}{\mathrm{a6}}{+}{2}{}{\mathrm{a2}}{,}{-}{\mathrm{a7}}{-}{2}{}{\mathrm{a5}}{,}{\mathrm{a7}}{+}{2}{}{\mathrm{a3}}{,}{-}{\mathrm{a8}}{+}{2}{}{\mathrm{a4}}{,}{\mathrm{a8}}{-}{2}{}{\mathrm{a12}}{,}{\mathrm{a9}}{-}{2}{}{\mathrm{a1}}{,}{\mathrm{a9}}{+}{2}{}{\mathrm{a11}}\right\}{,}\left[\left\{{\mathrm{a1}}{=}{0}{,}{\mathrm{a10}}{=}{0}{,}{\mathrm{a11}}{=}{0}{,}{\mathrm{a12}}{=}{0}{,}{\mathrm{a13}}{=}{0}{,}{\mathrm{a14}}{=}{0}{,}{\mathrm{a15}}{=}{0}{,}{\mathrm{a16}}{=}{0}{,}{\mathrm{a17}}{=}{0}{,}{\mathrm{a18}}{=}{0}{,}{\mathrm{a19}}{=}{0}{,}{\mathrm{a2}}{=}{0}{,}{\mathrm{a20}}{=}{0}{,}{\mathrm{a21}}{=}{0}{,}{\mathrm{a3}}{=}{0}{,}{\mathrm{a4}}{=}{0}{,}{\mathrm{a5}}{=}{0}{,}{\mathrm{a6}}{=}{0}{,}{\mathrm{a7}}{=}{0}{,}{\mathrm{a8}}{=}{0}{,}{\mathrm{a9}}{=}{0}\right\}\right]{,}\left[\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}{,}{\mathrm{e6}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}{,}{\mathrm{e5}}{,}{\mathrm{e7}}{,}{\mathrm{e8}}{,}{\mathrm{e9}}{,}{\mathrm{e10}}\right]\right]\right]$ (6.13)
 sp2 > ReductivePairs[1];
 $\left[\left[{\mathrm{e2}}{,}{\mathrm{e4}}{,}{\mathrm{e6}}\right]{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}{,}{\mathrm{e5}}{,}{\mathrm{e7}}{,}{\mathrm{e8}}{,}{\mathrm{e9}}{,}{\mathrm{e10}}\right]\right]$ (6.14)

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

 sp2 > Query(op(ReductivePairs[1]),"SymmetricPair");
 ${\mathrm{false}}$ (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);
 ${\mathrm{\phi }}{≔}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx3}}{+}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx5}}{-}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx6}}{}{\bigwedge }{}{\mathrm{dx7}}{+}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx6}}{+}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx5}}{}{\bigwedge }{}{\mathrm{dx7}}{+}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx7}}{-}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx5}}{}{\bigwedge }{}{\mathrm{dx6}}$ (7.1)
 E7 > V:= D_x1;
 ${V}{≔}{\mathrm{D_x1}}$ (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]);
 ${\mathrm{g2_subalg}}{≔}\left[{\mathrm{e12}}{-}{\mathrm{e21}}{+}{\mathrm{e56}}{-}{\mathrm{e65}}{,}{\mathrm{e13}}{-}{\mathrm{e31}}{+}{\mathrm{e57}}{-}{\mathrm{e75}}{,}{\mathrm{e14}}{-}{\mathrm{e36}}{-}{\mathrm{e41}}{+}{\mathrm{e63}}{,}{\mathrm{e15}}{-}{\mathrm{e37}}{-}{\mathrm{e51}}{+}{\mathrm{e73}}{,}{\mathrm{e16}}{+}{\mathrm{e34}}{-}{\mathrm{e43}}{-}{\mathrm{e61}}{,}{\mathrm{e17}}{+}{\mathrm{e35}}{-}{\mathrm{e53}}{-}{\mathrm{e71}}{,}{\mathrm{e23}}{-}{\mathrm{e32}}{+}{\mathrm{e67}}{-}{\mathrm{e76}}{,}{\mathrm{e24}}{+}{\mathrm{e35}}{-}{\mathrm{e42}}{-}{\mathrm{e53}}{,}{\mathrm{e25}}{-}{\mathrm{e34}}{+}{\mathrm{e43}}{-}{\mathrm{e52}}{,}{\mathrm{e26}}{-}{\mathrm{e37}}{-}{\mathrm{e62}}{+}{\mathrm{e73}}{,}{\mathrm{e27}}{+}{\mathrm{e36}}{-}{\mathrm{e63}}{-}{\mathrm{e72}}{,}{\mathrm{e45}}{-}{\mathrm{e54}}{+}{\mathrm{e67}}{-}{\mathrm{e76}}{,}{\mathrm{e46}}{-}{\mathrm{e57}}{-}{\mathrm{e64}}{+}{\mathrm{e75}}{,}{\mathrm{e47}}{+}{\mathrm{e56}}{-}{\mathrm{e65}}{-}{\mathrm{e74}}\right]$ (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}{,}{\mathrm{h0}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e5}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e10}}{+}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e10}}{+}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e5}}{+}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{2}{}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{2}{}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e11}}{-}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e5}}{+}{\mathrm{e9}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e12}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{2}{}{\mathrm{e13}}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e13}}{+}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{2}{}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e14}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e6}}{-}{\mathrm{e8}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e12}}{+}{\mathrm{e7}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e1}}{+}{\mathrm{e14}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e13}}{-}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e5}}{-}{\mathrm{e9}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e12}}{+}{\mathrm{e7}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{-}{2}{}{\mathrm{e11}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e11}}\right]{=}{2}{}{\mathrm{e10}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{-}{2}{}{\mathrm{e12}}{+}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e11}}\right]{=}{-}{2}{}{\mathrm{e7}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e13}}\right]{=}{-}{2}{}{\mathrm{e14}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e14}}\right]{=}{2}{}{\mathrm{e13}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e14}}\right]{=}{-}{2}{}{\mathrm{e12}}\right]{,}\left[\left[\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}\right]\right]\right]$ (7.4)
 gl7 > DGsetup(g);
 ${\mathrm{Lie algebra: g2}}$ (7.5)
 g2 > Fr := Tools:-DGinfo("FrameBaseVectors");
 ${\mathrm{Fr}}{≔}\left[{\mathrm{e1}}{,}{\mathrm{e2}}{,}{\mathrm{e3}}{,}{\mathrm{e4}}{,}{\mathrm{e5}}{,}{\mathrm{e6}}{,}{\mathrm{e7}}{,}{\mathrm{e8}}{,}{\mathrm{e9}}{,}{\mathrm{e10}}{,}{\mathrm{e11}}{,}{\mathrm{e12}}{,}{\mathrm{e13}}{,}{\mathrm{e14}}\right]$ (7.6)
 g2 > su3 := map(DGzip,h0[1], Fr, "plus");
 ${\mathrm{su3}}{≔}\left[{\mathrm{e7}}{,}{\mathrm{e8}}{,}{\mathrm{e9}}{,}{\mathrm{e10}}{,}{\mathrm{e11}}{,}{\mathrm{e12}}{,}{\mathrm{e13}}{,}{\mathrm{e14}}\right]$ (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);
 ${C}{≔}\left[\left[\right]{,}\left[\right]{,}\left[{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ4}}{+}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}\right]{,}\left[{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ6}}{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ6}}{+}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ5}}{,}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ6}}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ5}}{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ5}}{-}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ6}}\right]{,}\left[{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ4}}{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}{-}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}\right]{,}\left[\right]{,}\left[{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}\right]{,}\left[\right]\right]$ (7.8)

Calculate the Lie algebra cohomology of g2 relative to su3.

 g2 > H := Cohomology(C);
 ${H}{≔}\left[\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[\right]{,}\left[{-}{\mathrm{θ1}}{}{\bigwedge }{}{\mathrm{θ2}}{}{\bigwedge }{}{\mathrm{θ3}}{}{\bigwedge }{}{\mathrm{θ4}}{}{\bigwedge }{}{\mathrm{θ5}}{}{\bigwedge }{}{\mathrm{θ6}}\right]\right]$ (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);
 ${\mathrm{m0}}{≔}\left[{\mathrm{e1}}{+}{\mathrm{a1}}{}{\mathrm{e7}}{+}{\mathrm{a2}}{}{\mathrm{e8}}{+}{\mathrm{a3}}{}{\mathrm{e9}}{+}{\mathrm{a4}}{}{\mathrm{e10}}{+}{\mathrm{a5}}{}{\mathrm{e11}}{+}{\mathrm{a6}}{}{\mathrm{e12}}{+}{\mathrm{a7}}{}{\mathrm{e13}}{+}{\mathrm{a8}}{}{\mathrm{e14}}{,}{\mathrm{e2}}{+}{\mathrm{a9}}{}{\mathrm{e7}}{+}{\mathrm{a10}}{}{\mathrm{e8}}{+}{\mathrm{a11}}{}{\mathrm{e9}}{+}{\mathrm{a12}}{}{\mathrm{e10}}{+}{\mathrm{a13}}{}{\mathrm{e11}}{+}{\mathrm{a14}}{}{\mathrm{e12}}{+}{\mathrm{a15}}{}{\mathrm{e13}}{+}{\mathrm{a16}}{}{\mathrm{e14}}{,}{\mathrm{e3}}{+}{\mathrm{a17}}{}{\mathrm{e7}}{+}{\mathrm{a18}}{}{\mathrm{e8}}{+}{\mathrm{a19}}{}{\mathrm{e9}}{+}{\mathrm{a20}}{}{\mathrm{e10}}{+}{\mathrm{a21}}{}{\mathrm{e11}}{+}{\mathrm{a22}}{}{\mathrm{e12}}{+}{\mathrm{a23}}{}{\mathrm{e13}}{+}{\mathrm{a24}}{}{\mathrm{e14}}{,}{\mathrm{e4}}{+}{\mathrm{a25}}{}{\mathrm{e7}}{+}{\mathrm{a26}}{}{\mathrm{e8}}{+}{\mathrm{a27}}{}{\mathrm{e9}}{+}{\mathrm{a28}}{}{\mathrm{e10}}{+}{\mathrm{a29}}{}{\mathrm{e11}}{+}{\mathrm{a30}}{}{\mathrm{e12}}{+}{\mathrm{a31}}{}{\mathrm{e13}}{+}{\mathrm{a32}}{}{\mathrm{e14}}{,}{\mathrm{e5}}{+}{\mathrm{a33}}{}{\mathrm{e7}}{+}{\mathrm{a34}}{}{\mathrm{e8}}{+}{\mathrm{a35}}{}{\mathrm{e9}}{+}{\mathrm{a36}}{}{\mathrm{e10}}{+}{\mathrm{a37}}{}{\mathrm{e11}}{+}{\mathrm{a38}}{}{\mathrm{e12}}{+}{\mathrm{a39}}{}{\mathrm{e13}}{+}{\mathrm{a40}}{}{\mathrm{e14}}{,}{\mathrm{e6}}{+}{\mathrm{a41}}{}{\mathrm{e7}}{+}{\mathrm{a42}}{}{\mathrm{e8}}{+}{\mathrm{a43}}{}{\mathrm{e9}}{+}{\mathrm{a44}}{}{\mathrm{e10}}{+}{\mathrm{a45}}{}{\mathrm{e11}}{+}{\mathrm{a46}}{}{\mathrm{e12}}{+}{\mathrm{a47}}{}{\mathrm{e13}}{+}{\mathrm{a48}}{}{\mathrm{e14}}\right]{,}\left\{{\mathrm{a1}}{,}{\mathrm{a10}}{,}{\mathrm{a11}}{,}{\mathrm{a12}}{,}{\mathrm{a13}}{,}{\mathrm{a14}}{,}{\mathrm{a15}}{,}{\mathrm{a16}}{,}{\mathrm{a17}}{,}{\mathrm{a18}}{,}{\mathrm{a19}}{,}{\mathrm{a2}}{,}{\mathrm{a20}}{,}{\mathrm{a21}}{,}{\mathrm{a22}}{,}{\mathrm{a23}}{,}{\mathrm{a24}}{,}{\mathrm{a25}}{,}{\mathrm{a26}}{,}{\mathrm{a27}}{,}{\mathrm{a28}}{,}{\mathrm{a29}}{,}{\mathrm{a3}}{,}{\mathrm{a30}}{,}{\mathrm{a31}}{,}{\mathrm{a32}}{,}{\mathrm{a33}}{,}{\mathrm{a34}}{,}{\mathrm{a35}}{,}{\mathrm{a36}}{,}{\mathrm{a37}}{,}{\mathrm{a38}}{,}{\mathrm{a39}}{,}{\mathrm{a4}}{,}{\mathrm{a40}}{,}{\mathrm{a41}}{,}{\mathrm{a42}}{,}{\mathrm{a43}}{,}{\mathrm{a44}}{,}{\mathrm{a45}}{,}{\mathrm{a46}}{,}{\mathrm{a47}}{,}{\mathrm{a48}}{,}{\mathrm{a5}}{,}{\mathrm{a6}}{,}{\mathrm{a7}}{,}{\mathrm{a8}}{,}{\mathrm{a9}}\right\}$ (7.10)
 g2 > TF, Eq, Soln, ReductivePairs:= Query(su3,m0,"ReductivePair"):
 g2 > ReductivePairs;
 $\left[\left[\left[{\mathrm{e7}}{,}{\mathrm{e8}}{,}{\mathrm{e9}}{,}{\mathrm{e10}}{,}{\mathrm{e11}}{,}{\mathrm{e12}}{,}{\mathrm{e13}}{,}{\mathrm{e14}}\right]{,}\left[{\mathrm{e1}}{-}\frac{{\mathrm{e14}}}{{2}}{,}{\mathrm{e2}}{+}\frac{{\mathrm{e13}}}{{2}}{,}{\mathrm{e3}}{+}\frac{{\mathrm{e11}}}{{2}}{,}{\mathrm{e4}}{-}\frac{{\mathrm{e10}}}{{2}}{,}{\mathrm{e5}}{+}\frac{{\mathrm{e9}}}{{2}}{,}{\mathrm{e6}}{-}\frac{{\mathrm{e8}}}{{2}}\right]\right]\right]$ (7.11)
 g2 > Query(op(ReductivePairs[1]),"SymmetricPair");
 ${\mathrm{false}}$ (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);
 ${\mathrm{\phi }}{≔}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx3}}{+}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx5}}{-}{\mathrm{dx1}}{}{\bigwedge }{}{\mathrm{dx6}}{}{\bigwedge }{}{\mathrm{dx7}}{+}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx6}}{+}{\mathrm{dx2}}{}{\bigwedge }{}{\mathrm{dx5}}{}{\bigwedge }{}{\mathrm{dx7}}{+}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx4}}{}{\bigwedge }{}{\mathrm{dx7}}{-}{\mathrm{dx3}}{}{\bigwedge }{}{\mathrm{dx5}}{}{\bigwedge }{}{\mathrm{dx6}}$ (8.1)
 E7 > g := CanonicalTensors("Metric","bas",7,0):
 E7 > DGsetup(MatrixAlgebras("Full", 7, gl7R));
 ${\mathrm{Lie algebra: gl7R}}$ (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}{,}{\mathrm{h0}}{≔}\left[\left[{\mathrm{e1}}{,}{\mathrm{e2}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e11}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e7}}\right]{=}{\mathrm{e2}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e8}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e9}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e10}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e1}}{,}{\mathrm{e11}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e3}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e7}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e3}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e2}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e4}}\right]{=}{-}{\mathrm{e16}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e17}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e18}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e4}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e3}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e5}}\right]{=}{-}{\mathrm{e19}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e20}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e19}}\right]{=}{\mathrm{e5}}{,}\left[{\mathrm{e4}}{,}{\mathrm{e20}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e6}}\right]{=}{-}{\mathrm{e21}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e17}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e5}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e6}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e1}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e2}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e18}}\right]{=}{-}{\mathrm{e3}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e4}}{,}\left[{\mathrm{e6}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e5}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e8}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e15}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e12}}\right]{=}{\mathrm{e8}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e13}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e14}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e7}}{,}{\mathrm{e15}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e9}}\right]{=}{-}{\mathrm{e16}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e17}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e18}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e12}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e9}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e8}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e10}}\right]{=}{-}{\mathrm{e19}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e20}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e19}}\right]{=}{\mathrm{e10}}{,}\left[{\mathrm{e9}}{,}{\mathrm{e20}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e11}}\right]{=}{-}{\mathrm{e21}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e17}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e10}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e11}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e7}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e18}}\right]{=}{-}{\mathrm{e8}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e9}}{,}\left[{\mathrm{e11}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e10}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e13}}\right]{=}{-}{\mathrm{e16}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e17}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e18}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e16}}\right]{=}{\mathrm{e13}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e17}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e12}}{,}{\mathrm{e18}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e14}}\right]{=}{-}{\mathrm{e19}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e20}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e16}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e19}}\right]{=}{\mathrm{e14}}{,}\left[{\mathrm{e13}}{,}{\mathrm{e20}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e15}}\right]{=}{-}{\mathrm{e21}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e17}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e14}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e15}}{,}\left[{\mathrm{e15}}{,}{\mathrm{e18}}\right]{=}{-}{\mathrm{e12}}{,}\left[{\mathrm{e15}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e13}}{,}\left[{\mathrm{e15}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e14}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e17}}\right]{=}{-}{\mathrm{e19}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e18}}\right]{=}{-}{\mathrm{e20}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e19}}\right]{=}{\mathrm{e17}}{,}\left[{\mathrm{e16}}{,}{\mathrm{e20}}\right]{=}{\mathrm{e18}}{,}\left[{\mathrm{e17}}{,}{\mathrm{e18}}\right]{=}{-}{\mathrm{e21}}{,}\left[{\mathrm{e17}}{,}{\mathrm{e19}}\right]{=}{-}{\mathrm{e16}}{,}\left[{\mathrm{e17}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e18}}{,}\left[{\mathrm{e18}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e16}}{,}\left[{\mathrm{e18}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e17}}{,}\left[{\mathrm{e19}}{,}{\mathrm{e20}}\right]{=}{-}{\mathrm{e21}}{,}\left[{\mathrm{e19}}{,}{\mathrm{e21}}\right]{=}{\mathrm{e20}}{,}\left[{\mathrm{e20}}{,}{\mathrm{e21}}\right]{=}{-}{\mathrm{e19}}\right]{,}\left[\left[\left[{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}\right]{,}\left[{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}\right]{,}\left[{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{-1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{-1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}\right]{,}\left[{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{1}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}{0}{,}\right]\right]\right]$