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GradedLieAlgebras :: derLie

derLie -- constructing a graded derivation

Synopsis

Description

The generators of M=f.sourceLie are mapped to the elements in the last argument defs and they should be given as generalExpressionLie in L=f.targetLie. If no f of class MapLie is given, then the current Lie algebra L is used and the derivation d maps L to L (and f is the identity map). The set of elements of class DerLie is a Lie algebra with Lie multiplication multDerLie, however it does not belong to LieAlgebra if we do not have a finite presentation. It is checked by the program that d maps the relations in d.sourceLie to zero.

i1 : L=lieAlgebra({x,y},{},genSigns=>1)

o1 = L

o1 : LieAlgebra
i2 : M=lieAlgebra({a,b},{[b,a,b]},genSigns=>0,genWeights=>{2,2})

o2 = M

o2 : LieAlgebra
i3 : f = mapLie(L,M,{[x,x],[]})

o3 = f

o3 : MapLie
i4 : d1 = derLie(f,{[x,x],[x,y]})

o4 = d1

o4 : DerLie
i5 : peek d1

o5 = DerLie{[b, a, b] => []    }
            a => [x, x]
            b => [x, y]
            maplie => f
            signDer => 0
            sourceLie => M
            targetLie => L
            weightDer => {0, 0}
i6 : evalDerLie(d1,[a,a,b])

o6 = {{-2}, {[x, x, x, y, x, x]}}

o6 : List
i7 : useLie L

o7 = L

o7 : LieAlgebra
i8 : d2 = derLie({[x,x,y],[x,x,y]})

o8 = d2

o8 : DerLie
i9 : peek d2

o9 = DerLie{maplie => MapLie{...4...}}
            signDer => 0
            sourceLie => L
            targetLie => L
            weightDer => {2, 0}
            x => [x, x, y]
            y => [x, x, y]
i10 : peek d2.maplie

o10 = MapLie{sourceLie => L}
             targetLie => L
             x => [x]
             y => [y]
i11 : evalDerLie(d2,[x,x,y])

          1  1
o11 = {{- -, -}, {[x, x, y, x, x], [y, x, y, x, x]}}
          2  2

o11 : List

See also

Ways to use derLie :