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

Jonas Hartwig
Assistant Professor of Mathematics
Iowa State University

“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 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,rk).
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”.

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