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:
-
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
|