Representations of Multiparameter Quantum Groups

We construct representations of the quantum algebras ~Uₐ ₐ (gl (n) ) and ~Uₐ ₐ (sl (n) ) ~ which are in duality with the multiparameter quantum groups ~GLₐ ₐ (n), ~SLₐ ₐ (n), ~ respectively. These objects depend on ~n (n-1) /2+1~ deformation parameters ~q, q₈₉ (1 i <j n) which is the maximal possible number in the case of GL (n). The representations are labelled by n-1 complex numbers ~rᵢ~ and are acting in the space of formal power series of ~n (n-1) /2~ non-commuting variables. These variables generate quantum flag manifolds of ~GLₐ ₐ (n), ~SLₐ ₐ (n). The case n=3 is treated in more detail.