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<p>Root vectors in quantum groups (of finite type) generalize to fused currents in quantum loop groups [J. Ding, S. Khoroshkin, <i>Transform. Groups</i>, <b>5</b>, 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, <i>Transform. Groups</i>, <b>5</b>, No. 2, 111–120 (2000), B. Enriquez, <i>J. Lie Theory</i>, <b>13</b>, No. 1, 21–64 (2003), and B. Feigin, A. Odesskii, <i>NATO Sci., Ser. II, Math. Phys. Chem.</i>, <b>35</b> (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, <i>J. reine und angew. Math.</i>, <b>470</b>, 51–88 (1996)] corresponding to the orientation of the ADE-type Dynkin diagram.</p>

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Fusion and Specialization for Type ADE Shuffle Algebras

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  • 发表日期：

    2025年12月20日

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  • Andrei Neguţ,
  • Alexander Tsymbaliuk

摘要

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.