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# image(LieDerivation,LieSubSpace) -- make the image of a Lie subspace under a Lie derivation

## Synopsis

• Function: image
• Usage:
I=image(d,S)
• Inputs:
• d, an instance of the type LieDerivation,
• S, an instance of the type LieSubSpace, an instance of type LieSubSpace, a Lie subspace of the source of $d$
• Outputs:
• I, an instance of the type LieSubSpace, an instance of LieSubSpace, the image $d(S)$, a Lie subspace of the target of $d$

## Description

If $d$ is a differential on a Lie algebra $L$ and $S$ is an ideal in $L$, then image(d,S) is of type LieSubAlgebra. Otherwise, image(d,S) is of type LieSubSpace.

 i1 : F=lieAlgebra({a,b,c,r3,r4,r42}, Weights => {{1,0},{1,0},{2,0},{3,1},{4,1},{4,2}},Signs=>{0,0,0,1,1,0}, LastWeightHomological=>true) o1 = F o1 : LieAlgebra i2 : D=differentialLieAlgebra{0_F,0_F,0_F,a c,a a c,r4 - a r3} o2 = D o2 : LieAlgebra i3 : S=lieIdeal{a r3} o3 = S o3 : FGLieIdeal i4 : d=differential D o4 = d o4 : LieDerivation i5 : T=image(d,S) o5 = T o5 : LieSubAlgebra i6 : basis(5,T) o6 = {(b a a c), (a a a c)} o6 : List