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LieDerivation -- the class of all Lie algebra derivations


Given a homomorphism of Lie algebras $f: M \ \to\ L$, one has the notion of a derivation $d: M \ \to\ L$ over $f$, and LieDerivation is the type representing such pairs $(d,\,f)$ ($f$ is the identity for the case of ordinary derivations from $L$ to $L$). The derivation law reads \break $d$ [x, y] = [$d$ x, $f$ y] ± [$f$ x, $d$ y], \break where the sign is determined by the sign of interchanging $d$ and $x$, i.e., the sign is plus if sign$(d)$=0 or sign$(x)$=0 and minus otherwise. An object of type LieDerivation need not be well defined as a map. Use isWellDefined(ZZ,LieDerivation) to check if the derivation is well defined.

i1 : L = lieAlgebra{a,b}

o1 = L

o1 : LieAlgebra
i2 : M = lieAlgebra{a,b,c}

o2 = M

o2 : LieAlgebra
i3 : f = map(L,M)

o3 = f

o3 : LieAlgebraMap
i4 : use L
i5 : der = lieDerivation(f,{a a b,b b a,a a b+b b a})

o5 = der

o5 : LieDerivation
i6 : describe der

o6 = 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
i7 : use M
i8 : der a c

o8 =  - (a a b a) + (b a b a)

o8 : L

See also

Functions and methods returning an object of class LieDerivation :

Methods that use an object of class LieDerivation :

For the programmer

The object LieDerivation is a type, with ancestor classes HashTable < Thing.