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