# RingElement LieDerivation -- multiplication of a field element and a derivation

## Synopsis

• Operator: SPACE
• Usage:
e = a d
• Inputs:
• a, , $a$ is an element in L#Field, where $L$ is the target of $d$
• d, an instance of the type LieDerivation,
• Outputs:

## Description

The symbol SPACE is used as notation for multiplication by scalars. The scalars belong to L#Field, which must be the same as M#Field, where $d: M\ \to\ L$. If the field is not QQ, then the scalars are of type RingElement. If the field is QQ, then the scalars are of type Number, see Number LieDerivation.

 i1 : F = toField(ZZ/7[x]/{x^2+1}) o1 = F o1 : PolynomialRing i2 : M = lieAlgebra({a,b},Field=>F) o2 = M o2 : LieAlgebra i3 : L = lieAlgebra({a,b},Field=>F) o3 = L o3 : LieAlgebra i4 : f = map(L,M,{x a,3 b}) o4 = f o4 : LieAlgebraMap i5 : d = lieDerivation(f,{-x b,-2 a}) o5 = d o5 : LieDerivation i6 : describe (3*x) d o6 = a => (3)b b => (x)a map => f sign => 0 weight => {0, 0} source => M target => L