Representations of Multiparameter Quantum Groups

We construct representations of the quantum algebras ~U_q q(gl(n)) and ~U_q q(sl(n))~ which are in duality with the multiparameter quantum groups ~GL_q q(n), ~SL_q q(n),~ respectively. These objects depend on ~$n(n-1)/2+1$~ deformation parameters ~q,qᵢⱼ (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 ~GL_q q(n), ~SL_q q(n). The case $n=3$ is treated in more detail.