LieAlgebraMap * LieDerivation -- composition of a homomorphism and a derivation

Synopsis

Operator: *
Usage:

h = g*d
Inputs:
- g, an instance of the type LieAlgebraMap, a homomorphism from $L$ to $N$
- d, an instance of the type LieDerivation, a derivation from $M$ to $L$
Outputs:
- h, an instance of the type LieDerivation, a derivation from $M$ to $N$

Description

The composition of maps $g*d$ is a derivation $M\ \to\ N$, with the composition $g*f$ defining the module structure of $N$ over $M$, where $f: M\ \to\ L$ defines the module structure of $L$ over $M$.

i1 : L = lieAlgebra{a,b}

o1 = L

o1 : LieAlgebra

i2 : d = lieDerivation{a a b,b b a}

o2 = d

o2 : LieDerivation

i3 : describe d

o3 = a =>  - (a b a)
     b => (b b a)
     map => id_L
     sign => 0
     weight => {2, 0}
     source => L
     target => L

i4 : N = lieAlgebra{a1,b1}

o4 = N

o4 : LieAlgebra

i5 : g = map(N,L,{b1,a1})

o5 = g

o5 : LieAlgebraMap

i6 : h = g*d

o6 = h

o6 : LieDerivation

i7 : describe h

o7 = a => (b1 b1 a1)
     b =>  - (a1 b1 a1)
     map => g
     sign => 0
     weight => {2, 0}
     source => L
     target => N