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GradedLieAlgebras :: isWellDefined(ZZ,LieDerivation)

isWellDefined(ZZ,LieDerivation) -- whether a Lie derivation is well defined

Synopsis

Description

It is checked that the derivation $(d,f): M \ \to\ L$ maps the ideal of relations in $M$ to 0 up to degree $n$. More precisely, if $M=F/I$ where $F$ is free, and $p$ is the projection $F$ \ \to\ $M$, then the derivation $(d*p,f*p): F \ \to\ L$ maps $I$ to 0 in degrees $\le\ n$. If $n$ is big enough and $I$ is a list, then it is possible to get the information "the derivation is well defined for all degrees".

i1 : F=lieAlgebra{a,b}

o1 = F

o1 : LieAlgebra
i2 : L=F/{a a a b,b b b a}

o2 = L

o2 : LieAlgebra
i3 : e=euler L

o3 = e

o3 : LieDerivation
i4 : isWellDefined(4,e)
the derivation is well defined for all degrees

o4 = true
i5 : d4=lieDerivation{0_L,a b a b a}
warning: the derivation might not be well defined, use isWellDefined

o5 = d4

o5 : LieDerivation
i6 : isWellDefined(4,d4)

o6 = false
i7 : d5=lieDerivation{0_L,b a b a b a}
warning: the derivation might not be well defined, use isWellDefined

o7 = d5

o7 : LieDerivation
i8 : isWellDefined(4,d5)
the derivation is well defined for all degrees

o8 = true
i9 : di=innerDerivation(a b a b a)

o9 = d5

o9 : LieDerivation
i10 : isWellDefined(4,di)
the derivation is well defined for all degrees

o10 = true
i11 : di===d5

o11 = true

See also