Representations of a noncommutative associative algebra related to quantum torus of rank three. (Q2505376)

Let \(A_q\) be the complex associative algebra generated by elements \(t_1^{\pm 1},\dots,t_n^{\pm 1},\partial_1,\dots,\partial_n\) subject to the defining relations \(t_it_j=q_{ij}t_jt_i\), \([\partial_i,\partial_j]=0\), \([\partial_i,t_j]=\delta_{ij}t_j\) for all \(i,j\). Here \(q_{ij}\) are nonzero complex numbers. In the case \(n=3\) there are found three series of irreducible representations of \(A_q\) on the spaces \(\mathbb{C}[t_1^{\pm 1},t_2^{\pm 1},t_3]\), \(\mathbb{C}[t_1^{\pm 1},t_2^{\pm 1},t_3^{\pm 1}]\), \(\mathbb{C}[t_1^{\pm 1},t_2^{\pm 1},t_3,(t_1-b)^{-1}]\) parameterized by points from \(\mathbb{C}^3\). In each case the problem of isomorphism of representations is solved.