The \(q\)-Difference Noether Problem for Complex Reflection Groups and Quantum OGZ Algebras

Abstract

For any complex reflection group \(G=G(m,p,n)\), we prove that the \(G\)-invariants of the division ring of fractions of the \(n\):th tensor power of the quantum plane is a quantum Weyl field and give explicit parameters for this quantum Weyl field. This shows that the \(q\)-Difference Noether Problem has a positive solution for such groups, generalizing previous work by Futorny and the author. Moreover, the new result is simultaneously a \(q\)-deformation of the classical commutative case, and of the Weyl algebra case recently obtained by Eshmatov et al. Secondly, we introduce a new family of algebras called quantum OGZ algebras. They are natural quantizations of the OGZ algebras introduced by Mazorchuk originating in the classical Gelfand-Tsetlin formulas. Special cases of quantum OGZ algebras include the quantized enveloping algebra of \(\mathfrak{gl}_n\) and quantized Heisenberg algebras. We show that any quantum OGZ algebra can be naturally realized as a Galois ring in the sense of Futorny-Ovsienko, with symmetry group being a direct product of complex reflection groups \(G(m,p,r_k)\). Finally, using these results we prove that the quantum OGZ algebras satisfy the quantum Gelfand-Kirillov conjecture by explicitly computing their division ring of fractions.