# image(LieDerivation) -- make the image of a Lie derivation

## Synopsis

• Function: image
• Usage:
I=image(d)
• Inputs:
• Outputs:
• I, an instance of the type LieSubSpace, an instance of type LieSubSpace, the image of $d$

## Description

If $d$ is a differential on a Lie algebra $L$, then image(d) (which is the same as the boundaries of $L$) is of type LieSubAlgebra. Otherwise, image(d) 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 : d=differential D o3 = d o3 : LieDerivation i4 : J=image(d) o4 = J o4 : LieSubAlgebra i5 : dims(6,J) o5 = | 0 0 1 2 5 11 | | 0 0 0 1 2 6 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | 6 6 o5 : Matrix ZZ <--- ZZ i6 : dims(6,boundaries D) o6 = | 0 0 1 2 5 11 | | 0 0 0 1 2 6 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | | 0 0 0 0 0 0 | 6 6 o6 : Matrix ZZ <--- ZZ