The Abdus Salam International Centre for Theoretical Physics Induced Representations of the Multiparameter Hopf

We construct induced representations of the multiparameter Hopf superalgebras Uuq(gl(m=n)) and Uuq(sl(m=n)). The rst superalgebra we constructed earlier as the dual of the multiparameter quantum deformation of the supergroup GL(m=n). The second superalgebra is a Hopf subalgebra of the rst for a special choice of the parameters. The representations are labelled by m+n integer numbers, respectively m+n 1 complex numbers, and act in the space of formal power series of (m + n)(m + n 1)=2 noncommuting variables, of which mn are odd and the rest are even. These variables generate a q-deformation of a ag supermanifold of the supergroup GL(m=n), respectively SL(m=n).