Derivation -- Derivation defined on a noncommutative algebra

Synopsis

Usage:

delta = derivation(A,output,sigma)
Inputs:
- A, a free algebra, or a FreeAlgebraQuotient.
- output, a list,
- sigma, a ring map,
Outputs:
- delta, an instance of the type Derivation,

Description

This function returns a Derivation object, which may be used to perform computations with (twisted) derivations in a noncommutative algebra. A linear map $\delta : A \to A$ is called a $\sigma$-derivation provided for all $x,y \in A$, one has $\delta(xy) = \delta(x)y + \sigma(x)\delta(y)$. Such maps are useful in defining many noncommutative algebras, including Ore extensions.

Below we give a simple example of a twisted derivation that is used to define the subalgebras appearing in Fomin and Procesi's work to describe Fomin-Kirillov algebras.

i1 : A = QQ<|x,y|>

o1 = A

o1 : FreeAlgebra

i2 : sigma = map(A,A,{y,x})

o2 = map (A, A, {y, x})

o2 : RingMap A <--- A

i3 : delta = derivation(A,{-x*y,y*x},sigma)

o3 = Derivation{generators => HashTable{x => -x*y}}
                                        y => y*x
                homomorphism => map (A, A, {y, x})
                imageCache => MutableHashTable{}
                matrix => | -xy yx |
                source => A

o3 : Derivation

i4 : delta y^2

o4 = x*y*x + y*x*y

o4 : A

Methods that use an object of class Derivation :

"Derivation RingElement"
"Derivation ZZ"
"oreExtension(Ring,RingMap,Derivation,RingElement)" -- see oreExtension -- Creates an Ore extension of a noncommutative ring
"oreExtension(Ring,RingMap,Derivation,Symbol)" -- see oreExtension -- Creates an Ore extension of a noncommutative ring
"oreIdeal(Ring,RingMap,Derivation,RingElement)" -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
"oreIdeal(Ring,RingMap,Derivation,Symbol)" -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring

For the programmer

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