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

## Synopsis

• Function: isWellDefined
• Usage:
b=isWellDefined(n,d)
• Inputs:
• Outputs:
• b, , true if $d$ is well defined up to degree $n$, false otherwise

## 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