Skew derivations of quantum tori and quantum spaces

Abstract

We determine the $\sigma$-derivations of quantum tori and quantum affine spaces for a toric automorphism $\sigma$. By standard results, every toric automorphism $\sigma$ of a quantum affine space $\mathcal{A}$ and every $\sigma$-derivation of $\mathcal{A}$ extend uniquely to the corresponding quantum torus $\mathcal{T}$. We shall see that, for a toric automorphism $\sigma$, every $\sigma$-derivation of $\mathcal{T}$ is a unique sum of an inner $\sigma$-derivation and a $\sigma$-derivation that is conjugate to a derivation and that the latter is non-zero only if $\sigma$ is an inner automorphism of $\mathcal{T}$. This is applied to determine the $\sigma$-derivations of $\mathcal{A}$ for a toric automorphism $\sigma$, generalizing results of Alev and Chamarie for the derivations of quantum affine spaces and of Almulhem and Brzezi\'{n}ski for $\sigma$-derivations of the quantum plane. We apply the results to iterated Ore extensions $A$ of the base field for which all the defining endomorphisms are automorphisms and each of the adjoined indeterminates is an eigenvector for all the subsequent defining automorphisms. We present an algorithm which, in characteristic zero, will, for such an algebra $A$, either construct a quantum torus between $A$ and its quotient division algebra or show that no such quantum torus exists. Also included is a general section on skew derivations which become inner on localization at the powers of a normal element which is an eigenvector for the relevant automorphism. This section explores a connection between such skew derivations and normalizing sequences of length two. This connection is illustrated by known examples of skew derivations and by the construction of a family of skew derivations for the parametric family of subalgebras of the Weyl algebra that has been studied in three papers by Benkart, Lopes and Ondrus.