Representations of Multiparameter Quantum Groups

Abstract

We construct representations of the quantum algebras Uq,q(gl(n)) and Uq,q(sl(n)) which are in duality with the multiparameter quantum groups GLqq(n), SLqq(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 GLqq(n), SLqq(n). The case n = 3 is treated in more detail.