An Elliptic Algebra $U_{q,p}(\hat{sl_2})$ and the Fusion RSOS Model

We introduce an elliptic algebra $U_{q,p}(\hat{sl_2})$ with $p=q^{2r} (r\in \R_{>0})$ and present its free boson representation at generic level $k$. We show that this algebra governs a structure of the space of states in the $k-$fusion RSOS model specified by a pair of positive integers $(r,k)$, or equivalently a $q-$deformation of the coset conformal field theory $SU(2)_k\times SU(2)_{r-k-2}/SU(2)_{r-2}$. Extending the work by Lukyanov and Pugai corresponding to the case $k=1$, we gives a full set of screening operators for $k>1$. The algebra $U_{q,p}(\hat{sl_2})$ has two interesting degeneration limits, $p\to 0$ and $p\to 1$. The former limit yields the quantum affine algebra $U_{q}(\hat{sl_2})$ whereas the latter yields the algebra ${\cal A}_{\hbar,\eta}(\hat{sl_2})$, the scaling limit of the elliptic algebra ${\cal A}_{q,p}(\hat{sl_2})$. Using this correspondence, we also obtain the highest component of two types of vertex operators which can be regarded as $q-$deformations of the primary fields in the coset conformal field theory.