A multiparameter family of irreducible representations of the quantum plane and of the quantum Weyl algebra. (Q2634626)

The quantum plane \(\mathbb F_q[x,y]\) is the associative algebra over a field \(\mathbb F\) generated by two elements \(x,y\) with defining relation \(xy-qyx=1\) where \(q\) is a non-unit nonzero element from \(\mathbb F\). Let \(f\) be a map \(\mathbb Z\to\mathbb F^*\) such that \(f(i+n)=qf(i)\) where \(n\) is a given positive integer. For a positive integer \(m\) denote by \(V^{m,n}_f\) a representation of the quantum plane on the space \(\mathbb F[t^{\pm 1}]\) such that \(x\cdot t^i=t^{i+n}\), and \(y\cdot t^i=f(i)t^{i-m}\). It is shown that \(V^{m,n}_f\) is irreducible if and only if \(n,m\) are coprime. If \(V^{m,n}_f\) and \(V^{m',n'}_g\) are isomorphic then \((n,m)=(n',m')\). A representation \(V^{m,n}_f\) is semisimple and a polynomial algebra in the variable \(H=xy\) (a weight representation) if and only if \(n=m\). Moreover if \(\mathbb F\) is algebraically closed and \(V\) is an irreducible infinite dimensional weight representation then \(V\) is isomorphic to \(V^{1,1}_f\).