The authors classify simple modules for a class of graded rings of the form \(R=\bigoplus_{i\in\mathbb{Z}}R_i\) where \(R_0=D\) is a commutative Dedekind domain, each \(R_i=Dv_i\) is a free left \(D\)-module of rank one, \(v_0=1\) and, for \(a,b\in D\), \(av_ibv_j=a\sigma^i(b)c(i,j)v_{i+j}\) for an automorphism \(\sigma\) of \(D\) and a \(2\)-cocycle \(c\). This class includes the generalized Weyl algebra or GWA \(D(\sigma,a)\) where \(0\neq a\in D\) and \(\sigma\) is an automorphism of \(D\). This is the ring extension of \(D\) generated by \(X\) and \(Y\) subject to the relations \(XY=a\), \(YX=\sigma(a)\) and, for all \(d\in D\), \(Xd=\sigma(d)X\) and \(Yd=\sigma^{-1}(d)Y\). In the above notation, \(v_i=X^i\) if \(i>0\) and \(v_i=Y^{-i}\) if \(i<0\).
The first author [Ukr. Mat. Zh. 44, No. 12, 1628-1644 (1992; Zbl 0810.16003), CMS Conf. Proc. 14, 83-107 (1993; Zbl 0806.17023) and Algebra Anal. 4, No. 1, 75-97 (1992; Zbl 0807.16027)], classified the simple modules of \(D(\sigma,a)\) in the case where no maximal ideal of \(D\) is invariant under a positive power of \(\sigma\) and the present paper generalizes this classification. One example, to which a section is devoted, is the quantum Weyl algebra \(A_1(q)\) over a field \(k\), which is the GWA \(D(\sigma,H)\) where \(\sigma(H)=q^{-1}(H-1)\). There is a \(\sigma\)-invariant maximal ideal of \(D\) which, if \(q\) is not a root of unity, is the unique maximal ideal of \(D\) invariant under a positive power of \(\sigma\).
As in the previous work of the first author and in earlier work of R. E. Block [Lect. Notes Math. 740, 69-79 (1979; Zbl 0414.16017)], on the Weyl algebra \(A_1\), the classification is in terms of the irreducible elements of the Euclidean ring \(B=K[X^{\pm 1};\sigma]\), where \(K\) is the quotient field of \(D\). The ring \(B\) is obtained from \(R\) by localization at \(D\setminus\{0\}\). The simple \(R\)-modules which are torsion with respect to \(D\setminus\{0\}\) are analysed separately from those which are torsion-free. It is shown that there are no torsion-free simple modules if and only if there are infinitely many maximal ideals of \(D\) invariant under positive powers of \(\sigma\).
One interesting example to which the results are applied and which is not a GWA is a factor \(V\) of the enveloping algebra of the Virasoro-Lie algebra. Here \(D=k[H]\), \(\sigma(H)=H-1\), \(v_iv_j=(H-i-1)v_{i+j}\) if \(i+j\neq 0\) and \(v_{-i}v_i=(H+i-1)(H-1)\).
The authors also consider the positively graded case and give a similar classification of simple modules, with the skew polynomial ring \(K[X;\sigma]\) replacing the skew Laurent polynomial ring \(B\) and with some modifications described as not entirely innocent. An example is given to show that the above result on the existence of torsion-free simple modules no longer holds.