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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 L=f.targetLie given in the last argument defs. It is checked by the program that d maps the relations in d.sourceLie to zero. 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). In this latter case, the set of elements of class DerLie is a Lie algebra with Lie multiplication DerLie DerLie. If L has a differential δ, then DerLie is a differential Lie algebra with differential d->[δ,d]. However DerLie does not belong to LieAlgebra unless a positively graded finite presentation can be given.

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

o1 = L

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

o2 = M

o2 : LieAlgebra
i3 : f = mapLie(L,M,{x x,L.zz})

o3 = f

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

o4 = d1

o4 : DerLie
i5 : peekLie d1

o5 = a => x
     b => y
     maplie => MapLie{a => (x x)    }
                      b => 0
                      sourceLie => M
                      targetLie => L
     sign => 1
     weight => {-1, 0}
     sourceLie => M
     targetLie => L
i6 : d1 a b

o6 =  - (y x x)

o6 : L
i7 : useLie L

o7 = L

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

o8 = d2

o8 : DerLie
i9 : peekLie d2

o9 = x => x
     y => y
     maplie => id
     sign => 0
     weight => {0, 0}
     sourceLie => L
     targetLie => L
i10 : d2 y y x

o10 = 3 (y y x)

o10 : L

See also

Ways to use derLie :