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LieAlgebraMap * LieAlgebraMap -- composition of homomorphisms

Synopsis

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

Description

If $f$ or $g$ is not well defined, then it may happen that $f*g$ is well defined and not equal to the map $x \ \to\ f(g(x))$ but equal to the map induced by the values $f(g(gen))$ where $gen$ is a generator for $M$. Here is an example of this fact for rings.

 i1 : R = QQ[x] o1 = R o1 : PolynomialRing i2 : S = R/(x*x) o2 = S o2 : QuotientRing i3 : f = map(R,S) o3 = map (R, S, {x}) o3 : RingMap R <--- S i4 : g = map(S,R) o4 = map (S, R, {x}) o4 : RingMap S <--- R i5 : h = f*g o5 = map (R, R, {x}) o5 : RingMap R <--- R i6 : isWellDefined f o6 = false i7 : isWellDefined h o7 = true i8 : use R o8 = R o8 : PolynomialRing i9 : h(x*x) 2 o9 = x o9 : R i10 : f(g(x*x)) o10 = 0 o10 : R
 i11 : L = lieAlgebra{a,b} o11 = L o11 : LieAlgebra i12 : f = map(L,L,{b,-a}) o12 = f o12 : LieAlgebraMap i13 : describe(f*f+id_L) o13 = a => 0 b => 0 source => L target => L