The optional input given above is not relevant for Lie algebras. If $d$ commutes with the differentials in the source and target of $d$, then the output is of type LieSubAlgebra. Otherwise, the output is of type LieSubSpace.
i1 : L = lieAlgebra({a,b,c},Weights=>{{1,0},{2,1},{3,2}},
Signs=>{1,1,1},LastWeightHomological=>true)
o1 = L
o1 : LieAlgebra
|
i2 : D= differentialLieAlgebra({0_L,a a,a b})
o2 = D
o2 : LieAlgebra
|
i3 : Q=D/{b b+4 a c}
o3 = Q
o3 : LieAlgebra
|
i4 : d=differential Q o4 = d o4 : LieDerivation |
i5 : Z=kernel d o5 = Z o5 : LieSubAlgebra |
i6 : C=cycles Q o6 = C o6 : LieSubAlgebra |
i7 : dims(8,Z)
o7 = | 1 1 0 0 0 0 0 0 |
| 0 0 1 1 1 1 1 1 |
| 0 0 0 0 0 0 1 2 |
| 0 0 0 0 1 1 1 1 |
| 0 0 0 0 0 0 0 1 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
8 8
o7 : Matrix ZZ <--- ZZ
|
i8 : dims(8,C)
o8 = | 1 1 0 0 0 0 0 0 |
| 0 0 1 1 1 1 1 1 |
| 0 0 0 0 0 0 1 2 |
| 0 0 0 0 1 1 1 1 |
| 0 0 0 0 0 0 0 1 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
| 0 0 0 0 0 0 0 0 |
8 8
o8 : Matrix ZZ <--- ZZ
|