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.