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

Synopsis

Function: inverse
Usage:

I=inverse(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 target of $f$
Outputs:
- I, an instance of the type LieSubSpace, an instance of LieSubSpace, a Lie subspace of the source of $f$, the inverse image of $S$ under $f$,

Description

If $S$ is an instance of LieIdeal, then $I$ is of type LieIdeal. If $S$ is an instance of LieSubAlgebra but not of LieIdeal, then $I$ is of type LieSubAlgebra. Otherwise, $I$ is of type LieSubSpace.

i1 : F=lieAlgebra{a,b,c}

o1 = F

o1 : LieAlgebra

i2 : I=lieIdeal{b c - a c}

o2 = I

o2 : FGLieIdeal

i3 : Q=F/I

o3 = Q

o3 : LieAlgebra

i4 : f=map(Q,F)

o4 = f

o4 : LieAlgebraMap

i5 : J=lieIdeal{a b}

o5 = J

o5 : FGLieIdeal

i6 : K=inverse(f,J)

o6 = K

o6 : LieIdeal

i7 : dims(1,6,F/K)

o7 = {3, 1, 2, 3, 6, 9}

o7 : List

i8 : dims(1,6,Q/J)

o8 = {3, 1, 2, 3, 6, 9}

o8 : List

Ways to use this method: