Quantum Linear Galois Algebras

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.