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GradedLieAlgebras :: lieDerivation

lieDerivation -- make a graded derivation

Description

Let f: M -> L be a map of Lie algebras. Let F be a free Lie algebra together with a surjective homomorphism p: F -> M. Define g: F -> L as the composition g=f*p. A derivation dF:F -> L over g is defined by defining dF on the generators of F and then extending dF to all of F by the derivation rule dF [x, y] = [dF x, g y] ± [g x, dF y], where the sign is plus if sign(d)=0 or sign(x)=0 and minus otherwise. The output d represents the induced map M -> L, which might not be well defined. That the derivation is indeed well defined may be checked (up to a certain degree) using isWellDefined(ZZ,LieDerivation). When no f of class LieAlgebraMap is given as input, the derivation d maps L to L (and f is the identity map). In this case, the set D of elements of class LieDerivation is a graded Lie algebra with Lie multiplication using SPACE. If L has differential del, then D is a differential Lie algebra with differential d -> [del,d]. If e is the Euler derivation on L, then d -> [e,d] is the Euler derivation on D.

Synopsis

  • Usage:
    d=lieDerivation(f,defs)
  • Inputs:
    • f, an instance of the type LieAlgebraMap
    • defs, a list, the values of the generators
  • Outputs:
i1 : L=lieAlgebra({x,y},Signs=>1)

o1 = L

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

o2 = M

o2 : LieAlgebra
i3 : f = map(L,M,{x x,0_L})
warning: the map might not be well defined, 
         use isWellDefined

o3 = f

o3 : LieAlgebraMap
i4 : d = lieDerivation(f,{x,y})
warning: the derivation might not be well defined, use isWellDefined

o4 = d

o4 : LieDerivation
i5 : isWellDefined(6,d)
the derivation is well defined for all degrees

o5 = true
i6 : describe d

o6 = a => x
     b => y
     map => f
     sign => 1
     weight => {-1, 0}
     source => M
     target => L
i7 : d a b

o7 =  - (y x x)

o7 : L

Synopsis

  • Usage:
    d=lieDerivation(defs)
  • Inputs:
    • defs, a list, the values of the generators
  • Outputs:
i8 : L=lieAlgebra({x,y},Signs=>1)

o8 = L

o8 : LieAlgebra
i9 : e = euler L

o9 = e

o9 : LieDerivation
i10 : d1 = lieDerivation{x y,0_L}

o10 = d1

o10 : LieDerivation
i11 : d3 = lieDerivation{x x x y,0_L}

o11 = d3

o11 : LieDerivation
i12 : describe d3

o12 = x =>  - (1/2)(x y x x)
      y => 0
      map => id_L
      sign => 1
      weight => {3, 0}
      source => L
      target => L
i13 : e d1

o13 = d1

o13 : LieDerivation
i14 : e d3

o14 = derivation from L to L

o14 : LieDerivation
i15 : oo===3 d3

o15 = true

See also

Ways to use lieDerivation :