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GradedLieAlgebras :: isWellDefined(ZZ,LieAlgebraMap)

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

Synopsis

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

See also