A Parametric Family of Subalgebras of the Weyl Algebra III. Derivations

TL;DR

This paper investigates the derivations of a family of subalgebras of the Weyl algebra, expanding understanding of their algebraic structure and symmetries over arbitrary fields.

Contribution

It provides a complete characterization of derivations for the algebra A_h, a significant extension of prior work on automorphisms and modules.

Findings

Derived explicit formulas for derivations of A_h.

Extended results to arbitrary fields.

Connected derivations to algebraic structure and symmetries.

Abstract

An Ore extension over a polynomial algebra F[x] is either a quantum plane, a quantum Weyl algebra, or an infinite-dimensional unital associative algebra A_h generated by elements x,y, which satisfy yx-xy = h, where h is in F[x]. When h is nonzero, these algebras are subalgebras of the Weyl algebra A_1 and can be viewed as differential operators with polynomial coefficients. In previous work, we investigated the structure of A_h, determined its automorphisms and their invariants, and studied the irreducible A_h-modules. Here we determine the derivations of A_h over an arbitrary field.