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

## Synopsis

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

## Description

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

 i1 : L = lieAlgebra{a,b} o1 = L o1 : LieAlgebra i2 : M = lieAlgebra{a,b,c} o2 = M o2 : LieAlgebra i3 : N = lieAlgebra{a1,b1} o3 = N o3 : LieAlgebra i4 : f = map(L,M) o4 = f o4 : LieAlgebraMap i5 : use M i6 : g = map(M,N,{b,a}) o6 = g o6 : LieAlgebraMap i7 : use L i8 : d = lieDerivation(f,{a a b,b b a,a a b+b b a}) o8 = d o8 : LieDerivation i9 : describe d o9 = a => - (a b a) b => (b b a) c => - (a b a) + (b b a) map => f sign => 0 weight => {2, 0} source => M target => L i10 : describe(f*g) o10 = a1 => b b1 => a source => N target => L i11 : describe(d*g) o11 = a1 => (b b a) b1 => - (a b a) map => homomorphism from N to L sign => 0 weight => {2, 0} source => N target => L