LieDerivation LieDerivation -- Lie multiplication of ordinary derivations

Synopsis

Operator: SPACE
Usage:

e = d1 d2
Inputs:
- d1, an instance of the type LieDerivation,
- d2, an instance of the type LieDerivation,
Outputs:
- e, an instance of the type LieDerivation, the Lie product of $d1$ and $d2$

Description

The vector space $D$ of graded derivations from $L$ to $L$ with the identity map as defining map, see LieDerivation, is a graded Lie algebra. If $L$ has a differential $del$, then $D$ is a differential graded Lie algebra with differential $d$\ \to\ [$del$,$d$].

i1 : L = lieAlgebra{a,b}/{a a a b,b b b a}

o1 = L

o1 : LieAlgebra

i2 : d0 = lieDerivation{a,b}

o2 = d0

o2 : LieDerivation

i3 : d2 = lieDerivation{a b a,0_L}
warning: the derivation might not be well defined, use isWellDefined

o3 = d2

o3 : LieDerivation

i4 : d4 = lieDerivation{a b a b a,0_L}
warning: the derivation might not be well defined, use isWellDefined

o4 = d4

o4 : LieDerivation

i5 : describe d2 d4

o5 = a => (a b a b a b a)
     b => 0
     map => id_L
     sign => 0
     weight => {6, 0}
     source => L
     target => L

i6 : describe d0 d4

o6 = a => 4 (a b a b a)
     b => 0
     map => id_L
     sign => 0
     weight => {4, 0}
     source => L
     target => L

Ways to use this method: