Macaulay2 » Documentation
Packages » AssociativeAlgebras :: Derivation
next | previous | forward | backward | up | index | toc

Derivation -- Derivation defined on a noncommutative algebra



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 :

For the programmer

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