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

Synopsis

Function: image
Usage:

I=image(d)
Inputs:
- d, an instance of the type LieDerivation,
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

Ways to use this method: