# kernel(LieDerivation) -- make the kernel of a map

## Description

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