ИСТИНА

For the quantum algebra U_q(gl(n+1)) in its reduction on the subalgebra U_q(gl(n)) an explicit description of a Mickelsson -- Zhelobenko reduction Z-algebra Z_q(gl(n+1);gl(n)) is given in terms of the generators and their defining relations. Using this Z-algebra we describe Hermitian irreducible representations of a discrete series for the quantum algebra U_q(u(n;1)) which is a real form of U_q(gl(n+1)), namely, an orthonormal Gelfand -- Graev basis is constructed in an explicit form and actions of the Uq(u(n;1))-generators in this basis are obtained. Next, we show as these results are generalized on the case U_q(u(n;2)).