# 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