Modules over quantum polynomials
The author uses methods from papers by \textit{J. T. Stafford} [J. Algebra 47, No. 2, 244-267 (1977; Zbl 0391.16009)], \textit{J. C. McConnell} and \textit{J. J. Pettit} [J. Lond. Math. Soc., II. Ser. 38, No. 1, 47-55 (1988; Zbl 0652.16007)], and the author [Mat. Sb. 185, No. 7, 3-12 (1994; Zbl 0849.16003)], to describe the structure of finitely generated modules over certain noncommutative algebras \(A_Q(n,n)\) of Laurent polynomials. The algebra \(A_Q(n,n)\) consists of polynomials in \(n\) variables \(X_i\), and their inverses, over a division algebra \(D\) over a field \(k\). The construction requires \(n\) \(k\)-automorphisms \(\alpha_i\) of \(D\) and \(n^2\) elements \(q_{ij}\in D^*\), \(1\leq i,j\leq n\), satisfying the equations \(q_{ii}=q_{ij}q_{ji}=Q_{ijr}Q_{jri}Q_{rij}=1\), where \(Q_{ijr}=q_{ij}\alpha_j(q_{ir})\). The following relations are satisfied: \(X_iz=\alpha_i(z)X_i\) for \(z\in D\) and \(1\leq i\leq n\); \(X_iX_j=q_{ij}X_jX_i\). A freeness condition is imposed on the \(q_{ij}\) under which \(A_Q(n,n)\) is a simple Noetherian hereditary domain. This leads to a structure theorem: if \(M\) is a finitely generated \(A_Q(n,n)\)-module of rank \(s\geq 0\) then \(M=t(M)\oplus N\) where the torsion module \(t(M)\) is cyclic Artinian, and \(N=0\) if \(s=0\), \(N\) is free if \(s\geq 2\) and \(N\) is isomorphic to a left ideal if \(n=1\). It is also established that every left ideal of \(A_Q(n,n)\) can be generated by two elements. The author also considers the algebra \(A_Q(r,n)\) constructed in the same way as \(A_Q(n,n)\), except that only \(r\) of the \(X_i\)'s are inverted. The main result in the case \(r<n\), where \(A_Q(r,n)\) is no longer simple, concerns Artinian modules. Such a module must be cyclic if \(r>0\) and, subject to an extra condition, the same is true when \(r=0\).
scientific article; zbMATH DE number 1028537