The 3-cyclic quantum Weyl algebras, their prime spectra and a classification of simple modules ($q$ is not a root of unity)

Abstract

The 3-cyclic quantum Weyl algebra $A=A(\alpha ,\beta ,\gamma ;q^2)$, where $\alpha ,\beta ,\gamma ,q^2\in \mathbb{K}$ (a ground field), is a quadratic Noetherian domain of Gelfand–Kirillov dimension 3 that is generated by three subalgebras each of them is either a quantum plane or the quantum Weyl algebra. For the algebras $A$, their prime, completely prime, primitive and maximal spectra are described together with containments of prime ideals (the Zariski–Jacobson topology on the spectrum) and simple $A$-modules are classified when $q^2$ is not a root of unity. For each prime ideal, an explicit set of ideal generators is given. The centre $Z(A)$ of $A$ is $\mathbb{K}[\Omega ]$ where $\Omega$ is a cubic element. A semisimplicity criterion for the category of finite dimensional $A$-modules is given. Criteria are presented for all ideals of the algebra $A$ to commute and for each ideal of $A$ to be a unique product of primes (up to order).