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# image(LieAlgebraMap,LieSubSpace) -- make the image of a Lie subspace under a Lie algebra map

## Synopsis

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

## Description

If $S$ is of type FGLieSubAlgebra, then image(f,S) is of type FGLieSubAlgebra. If $S$ is an instance of LieSubAlgebra, but not of FGLieSubAlgebra, then image(f,S) is of type LieSubAlgebra. Otherwise, image(f,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 : I=lieIdeal{b c - a c,a b,b r4 - a r4} o3 = I o3 : FGLieIdeal i4 : S=lieIdeal{a c} o4 = S o4 : FGLieIdeal i5 : Q=D/I o5 = Q o5 : LieAlgebra i6 : f=map(Q,D) o6 = f o6 : LieAlgebraMap i7 : T=image(f,S) o7 = T o7 : LieSubAlgebra i8 : basis(5,T) o8 = {(b b b c), (c b c)} o8 : List