Quantum linear Galois orders

Abstract

We define a class of quantum linear Galois algebras which include the universal enveloping algebra Uq(gln), the quantum Heisenberg Lie algebra and other quantum orthogonal Gelfand-Zetlin algebras of type A, the subalgebras of G-invariants of the quantum affine space, quantum torus for G = G(m, p, n), and of the quantum Weyl algebra for G = Sn. We show that all quantum linear Galois algebras satisfy the quantum Gelfand-Kirillov conjecture. Moreover, it is shown that the the subalgebras of invariants of the quantum affine space and of quantum torus for the reflection groups and of the quantum Weyl algebra for symmetric groups are, in fact, Galois orders over an adequate commutative subalgebras and free as right (left) modules over these subalgebras. In the rank 1 cases the results hold for an arbitrary finite group of automorphisms when the field is C.

Some properties of abstract Galois algebras and generalized Weyl algebras

“…Corollary 5.11 had appeared previously in [26], but our method of proof is different and elementary. We also generalize the result from [17,Theorem 8] and show that the invariants of the n-th quantized Weyl algebra under the action of G(m, p, n) are a Galois order as well.…”

“…In the fourth section we recall the notions of principal and rational Galois orders from [22], and we compare them with the notion of linear Galois algebras. We show that all the Galois orders in [18] and [17] are principal Galois orders (Theorem 4.9), and A Sn n in particular is also rational (Theorem 4.10), which raises the question of whether all Galois orders are principal. We finish by checking the validity of the Gelfand-Kirillov Conjecture for many Galois algebras (Theorem 4.11).…”

“…We notice that most known examples of Galois algebras are linear (cf. [15], [16], [11], [9], [18], [17] Proof. In both cases, we now that (L * M) is a domain, since it is isomorphic to an iterated Ore extension of L. Hence K = (L * M) G is an Ore domain by [10].…”

“…In this section we discuss the notion of principal and rational Galois orders introduced in [22], with relation to subrings of invariants, and show that all the Galois algebras in [18], [17] are principal Galois orders. We also compare the notion of rational Galois algebra with that of linear Galois algebra.…”

“…The original motivation of this enterprise is the Gelfand-Tsetlin theory of representations of gl n ( [20]) in the case of infinite dimensional modules ( [7]). This theory has led to breakthrough in representation theory for many algebras (see discussion in [11]); in particular U (gl n ) ( [16]) and its quantization ( [12]), finite W -algebras of type A ( [14]), OGZ and quantum OGZ algebras of type A ( [21], [22]), as well as their parabolic subalgebras ( [22]), an alternating analogue of U (gl n ) ( [23]), invariant subrings of rings of differential operators and quantum groups ( [17], [18]), quantized Coulomb branches ( [34]), and rational Cherednik algebras ( [26]). Other furthers aspects of the representation theory of Galois algebras were developed in [28], [33], [8], [13].…”