# isWellDefined(ZZ,LieAlgebraMap) -- whether a Lie map is well defined

## Synopsis

• Function: isWellDefined
• Usage:
b=isWellDefined(n,f)
• Inputs:
• Outputs:
• b, , true if $f$ is well defined and commutes with the differentials up to degree $n$, false otherwise

## Description

It is checked that the map $f: M \ \to\ L$ maps the relations in $M$ to 0 up to degree $n$ and that $f$ commutes with the differentials in $M$ and $L$. If $n$ is big enough and ideal(M) is of type List, then it is possible to get that $f$ maps all relations to 0, which is noted as the message "the map is well defined for all degrees". This may happen even if the map does not commute with the differential (see g in the example below).

 i1 : L=lieAlgebra({a,b},Signs=>1,LastWeightHomological=>true, Weights=>{{1,0},{2,1}}) o1 = L o1 : LieAlgebra i2 : F=lieAlgebra({a,b,c}, Weights=>{{1,0},{2,1},{5,2}},Signs=>1,LastWeightHomological=>true) o2 = F o2 : LieAlgebra i3 : D=differentialLieAlgebra{0_F,a a,a a a b} o3 = D o3 : LieAlgebra i4 : Q1=D/{a a a a b,a b a b + a c} o4 = Q1 o4 : LieAlgebra i5 : use F i6 : Q2=F/{a a a a b,a b a b + a c} o6 = Q2 o6 : LieAlgebra i7 : f=map(D,Q1) warning: the map might not be well defined, use isWellDefined o7 = f o7 : LieAlgebraMap i8 : isWellDefined(6,f) the map is not well defined the map commutes with the differential for all degrees o8 = false i9 : g=map(Q1,Q2) warning: the map might not be well defined, use isWellDefined o9 = g o9 : LieAlgebraMap i10 : isWellDefined(6,g) the map is well defined for all degrees the map does not commute with the differential o10 = false i11 : h=map(Q1,D) o11 = h o11 : LieAlgebraMap i12 : isWellDefined(6,h) the map is well defined for all degrees the map commutes with the differential for all degrees o12 = true