# inverse(LieDerivation,LieSubSpace) -- make the inverse image of a Lie subspace under a Lie derivation

## Synopsis

• Function: inverse
• Usage:
I=inverse(d,S)
• Inputs:
• d, an instance of the type LieDerivation,
• S, an instance of the type LieSubSpace, an instance of type LieSubSpace, a Lie subspace of the target of $d$
• Outputs:
• I, an instance of the type LieSubSpace, a Lie subspace of the source of $d$, the inverse image of $S$ under $d$, an instance of LieSubSpace

## Description

If $S$ is an instance of LieIdeal, then $I$ is of type LieSubAlgebra. Otherwise, $I$ is of type LieSubSpace.

 i1 : F = lieAlgebra({a,b,c},Weights=>{{1,0},{2,1},{3,2}}, Signs=>{1,1,1},LastWeightHomological=>true) o1 = F o1 : LieAlgebra i2 : D = differentialLieAlgebra({0_F,a a,a b}) o2 = D o2 : LieAlgebra i3 : d = differential D o3 = d o3 : LieDerivation i4 : B = boundaries D o4 = B o4 : LieSubAlgebra i5 : x = (a a b a c) + (a a a b c) o5 = (a a b a c) + (a a a b c) o5 : D i6 : member(x,B) o6 = true i7 : S = inverse(d,lieIdeal{x}) o7 = S o7 : LieSubAlgebra i8 : weight x o8 = {8, 3} o8 : List i9 : basis(8,4,S) o9 = {(a a c c), (b b a c) + (b a b c)} o9 : List i10 : d\oo o10 = {2 (a a b a c) + 2 (a a a b c), 2 (a a b a c) + 2 (a a a b c)} o10 : List