Theq-difference Noether problem for complex reflection groups and quantum OGZ algebras

Abstract

Abstract. For any complex reflection group G = G(m, p, n), we prove that the Ginvariants 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 [10]. 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. [8].Secondly, we introduce a new family of algebras called quantum OGZ algebras. They are natural quantizations of the OGZ algebras introduced by Mazorchuk [18] 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 [11], 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.

Principal Galois orders and Gelfand-Zeitlin modules

“…. , n) when k = C, as shown in [FH14,H17], the proof of the conjecture is a direct consequence of Theorem 1.3. Theorem 1.4.…”

Quantum linear Galois orders

“…An algebra A is said to satisfy the quantum Gelfand-Kirillov conjecture if Frac(A) is isomorphic to a quantum Weyl field over a purely transcendental extension of k. We will say that two domains D 1 and D 2 are birationally equivalent if Frac(D 1 ) ≃ Frac(D 2 ). The quantum Gelfand-Kirillov conjecture is strongly connected with the q-difference Noether problem for reflection groups introduced in [17]. This problem asks whether the invariant quantum Weyl subfield (FracA q n (k)) W is isomorphic to some quantum Weyl field, where W is a reflection group.…”

“…This problem asks whether the invariant quantum Weyl subfield (FracA q n (k)) W is isomorphic to some quantum Weyl field, where W is a reflection group. The positive solution of the q-difference Noether problem was obtained in [17] for classical reflection groups. Using this fact, the validity of the quantum Gelfand-Kirillov conjecture was shown for the quantum universal enveloping algebra U q (gl n ) ( [10]) and for the quantum Orthogonal Gelfand-Zetlin algebras of type A ( [17]).…”

“…The positive solution of the q-difference Noether problem was obtained in [17] for classical reflection groups. Using this fact, the validity of the quantum Gelfand-Kirillov conjecture was shown for the quantum universal enveloping algebra U q (gl n ) ( [10]) and for the quantum Orthogonal Gelfand-Zetlin algebras of type A ( [17]). The latter class includes the simplyconnected quantized form of gl n , Ǔ (gl n ) and the quantized Heisenberg Liealgebra among the others.…”

First-order characterization of noncommutative birational equivalence

“…action that gives a positive solution to Noether's Problem gives a positive solution to noncommutative Noether's Problem as well ( [21]), with connections to constructive aspects of noncommutative invariant theory, and the modified Gelfand-Kirillov Conjecture discussed in [23]. Other noncommutative analogues of Noether's Problem have shown interesting applications in the study of skew field of fractions of quantum groups ( [19], [26]).…”