Fusion and Specialization for Type ADE Shuffle Algebras

Root vectors in quantum groups (of finite type) generalize to fused currents in quantum loop groups [J. Ding, S. Khoroshkin, Transform. Groups , 5 , No. 1, 35–59 (2000)]. In the present paper, we construct fused currents as duals to specialization maps of the corresponding shuffle algebras [B. Enriquez, Transform. Groups , 5 , No. 2, 111–120 (2000), B. Enriquez, J. Lie Theory , 13 , No. 1, 21–64 (2003), and B. Feigin, A. Odesskii, NATO Sci., Ser. II, Math. Phys. Chem. , 35 (2001)] in types ADE; an approach, which has a potential for generalization to arbitrary Kac–Moody types. Both root vectors and fused currents depend on a convex order of the positive roots and the choice we make in the present paper is that of the Auslander–Reiten order [C. Ringel, J. reine und angew. Math. , 470 , 51–88 (1996)] corresponding to the orientation of the ADE-type Dynkin diagram.