Fusion of \ade Lattice Models

Fusion hierarchies of \ade face models are constructed. The fused critical $D$, $E$ and elliptic $D$ models yield new solutions of the Yang-Baxter equations with bond variables on the edges of faces in addition to the spin variables on the corners. It is shown directly that the row transfer matrices of the fused models satisfy special functional equations. Intertwiners between the fused \ade models are constructed by fusing the cells that intertwine the elementary face weights. As an example, we calculate explicitly the fused $2\times 2$ face weights of the 3-state Potts model associated with the $D_4$ diagram as well as the fused intertwiner cells for the $A_5$--$D_4$ intertwiner. Remarkably, this $2\times 2$ fusion yields the face weights of both the Ising model and 3-state CSOS models.