Representations of Multiparameter Quantum Groups

Abstract

We construct representations of the quantum algebras ~$U_{q{\bf q}}(gl(n))$ and ~$U_{q{\bf q}}(sl(n))$~ which are in duality with the multiparameter quantum groups ~$GL_{q{\bf q}}(n)$, ~$SL_{q{\bf q}}(n)$,~ respectively. These objects depend on ~$n(n-1)/2+1$~ deformation parameters ~$q,q_{ij}$ ($1\leq i <j\leq n$) which is the maximal possible number in the case of $GL(n)$. The representations are labelled by $n-1$ complex numbers ~$r_i$~ 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 ~$GL_{q{\bf q}}(n)$, ~$SL_{q{\bf q}}(n)$. The case $n=3$ is treated in more detail.