Resumen de A tale of two shuffle algebras

Andrei Negut

• 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).