# 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