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# image(LieAlgebraMap) -- make the image of a Lie algebra map

- Function: image
- Usage:
    `I=image(f)`
- Inputs:
  - f, an instance of the type LieAlgebraMap,
- Outputs:
  - I, an instance of the type FGLieSubAlgebra, the image of $f$, a finitely generated Lie subalgebra of the target of $f$

## Description

    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 : Q=D/I

    o4 = Q

    o4 : LieAlgebra

    i5 : f=map(Q,D)

    o5 = f

    o5 : LieAlgebraMap

    i6 : T=image(f)

    o6 = T

    o6 : FGLieSubAlgebra

    i7 : isSurjective f

    o7 = true

## See also

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

## Ways to use this method:

- image(LieAlgebraMap) -- make the image of a Lie algebra map

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/GradedLieAlgebras/doc.m2:1233:0.