Relating the approaches to quantised algebras and quantum groups

This paper constructs two representations of the quantum groupU _q g' by exploiting its quotient structure and the quantum double construction. Here the quantum group is taken as the dual to the quantised algebraU _q g, a one parameter deformation of the universal enveloping algebra of the Lie algebra g, as in Drinfel'd 6] and Jimbo 10]. From the two representations, the Hopf structure of the quantised algebraU _q g is reexpressed in a matrix format. This is the very structure given by Faddeev et al. 7], in their approach to defining quantum groups and quantised algebras via the quantisation of the function space of the associated Lie group to g.Supported by a SERC studentship