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

- 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

## See also

- image(LieAlgebraMap,LieSubSpace) -- make the image of a Lie subspace under a Lie algebra map
- image(LieAlgebraMap) -- make the image of a Lie algebra map
- image(LieDerivation) -- make the image of a Lie derivation
- kernel(LieDerivation) -- make the kernel of a map
- inverse(LieDerivation,LieSubSpace) -- make the inverse image of a Lie subspace under a Lie derivation

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/doc.m2:1276:0.