Classifications of prime ideals and simple modules of the quantum Weyl algebra A1(q) (q is a root of unity)

This paper consists of three parts: (I) To develop general theory of a (large) class of central simple finite-dimensional algebras and answering some natural questions about them (that in general situation it is not even clear how to approach them, and the Brauer group is a step in the right directions), (II) to introduce and develop general theory of a large class of rings, PLM-rings, (intuitively, they are the most general form of Quillen’s Lemma), and (III) to apply these results for the following four classic algebras that turned out to be PLM-rings. Let [Formula: see text] be an arbitrary field and [Formula: see text] be a primitive [Formula: see text]th root of unity. Classifications of prime, completely prime, maximal and primitive ideals, and simple modules are obtained for the quantum Weyl algebra [Formula: see text], the skew polynomial algebra [Formula: see text], the skew Laurent polynomial algebras [Formula: see text], and [Formula: see text] where [Formula: see text]. The quotient rings (of fractions) of prime factor algebras of the algebras [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] are explicitly described. Each quotient ring is a central simple finite-dimensional algebra, i.e. isomorphic to the matrix algebra [Formula: see text] for some [Formula: see text] and a central simple division algebra [Formula: see text]. The division algebra [Formula: see text] is either a finite field extension of [Formula: see text] or a cyclic algebra. These descriptions are a key fact in the classifications of prime ideals, completely prime ideals, and simple modules for the algebras above. For each simple module its basis, dimension, and endomorphism algebra are given. Explicit descriptions are obtained for the automorphism groups of the algebras [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text].