Algebraic curves for commuting elements in the \(q\)-deformed Heisenberg algebra (Q1014600)

Let \(K\) be a field and let \(q\neq 0\) be an element of \(K\) such that \(q\) is not a root of unity if \(\text{char}(K)\neq 0\), and \(q\) is not a root of unity other than 1 if \(char(K)=0\). Let \(H_K(q)\) be the \(q\)-deformed Heisenberg algebra, which is generated by two elements \(A\) and \(B\), subject to the relation \(AB-qBA=1\). The authors extend the eliminant construction of \textit{J. L. Burchnall} and \textit{T. W. Chaundy} [Proc. Lond. Math. Soc. (2) 21, 420--440 (1923; JFM 49.0311.03)] and prove that two commuting elements of \(H_K(q)\) lie on a curve, providing also some information about the possible such curves.