  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.
