A tale of two shuffle algebras

• Andrei Negut ^[1]
  1. [1] Department of Mathematics, MIT, Cambridge, MA, USA Simion Stoilow Institute of Mathematics, Bucharest, Romania
• Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 30, Nº. 4, 2024, págs. 1-95
• Idioma: inglés
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• Resumen

  • As a quantum affinization, the quantum toroidal algebraUq,q (gl¨ n)is defined in terms of its “left” and “right” halves, which both admit shuffle algebra presentations (Enriquez in Transform Groups 5(2):111–120, 2000; Feigin and Odesskii in Am Math Soc Transl Ser 2:185, 1998). In the present paper, we take an orthogonal viewpoint, and give shuffle algebra presentations for the “top” and “bottom” halves instead, starting from the evaluation representation Uq (gl˙ n) Cn(z) and its usual R-matrix R(z) ∈ End(Cn⊗Cn)(z)(see Faddeev et al. in LeningradMath J 1:193–226, 1990). An upshot of this construction is a new topological coproduct on Uq,q (gl¨ n) which extends the Drinfeld–Jimbo coproduct on the horizontal subalgebra Uq (gl˙ n) ⊂ Uq,q (gl¨ n).