Primitive ideals in the algebra of regular functions on quantum \(m\times n\)-matrices (Q1810077)

Let \(q\) be a complex number which is not a root of 1 and \({\mathcal O}_q(M(m\times n,\mathbb{C}))\) the coordinate complex algebra of functions on rectangular \(m\times n\) matrices where \(m\leqslant n\). A Poisson bracket associated with \({\mathcal O}_q(M(m\times n,\mathbb{C}))\) is introduced on the commutative algebra \({\mathcal O}(M(m\times n,\mathbb{C}))={\mathcal O}_1(M(m\times n,\mathbb{C}))\). It is shown that the set of primitive ideals in \({\mathcal O}_q(M(m\times n,\mathbb{C}))\) (and symplectic leaves of \({\mathcal O}(M(m\times n,\mathbb{C}))\)) are disjoint unions of subsets \(\text{Prim}_w\) (\(\text{Symp}_w\), respectively) associated with elements \(w\in S_m\times S_m\), where \(S_m\) is the symmetric group. The main result of the paper shows the existence of a bijection \(\beta\colon\text{Prim}_w\to\text{Symp}_w\) such that \(\dim\beta(P)=\text{GKdim }{\mathcal O}_q(M(m\times n,\mathbb{C}))/P\) for any \(P\in\text{Prim}_w\). Similar problems for the prime spectrum and any field in the case \(n=m\) were considered by \textit{G. Cauchon} [J. Algebra 260, No. 2, 519-569 (2003; Zbl 1024.16001)].