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 GelfandTsetlin 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 FutornyOvsienko, 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 GelfandKirillov conjecture by explicitly computing their division ring of fractions.
 Publication:

arXiv eprints
 Pub Date:
 December 2015
 arXiv:
 arXiv:1512.09234
 Bibcode:
 2015arXiv151209234H
 Keywords:

 Mathematics  Quantum Algebra;
 17B37 (Primary);
 16K40;
 16S35 (Secondary)
 EPrint:
 13 pages. v2: minor corrections