# LieDerivation LieDerivation -- Lie multiplication of ordinary derivations

## Synopsis

• Operator: SPACE
• Usage:
e = d1 d2
• Inputs:
• 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