A tale of two shuffle algebras

As a quantum affinization, the quantum toroidal algebra \({U_{q,{{\overline{q}}}}(\ddot{{\mathfrak {gl}}}_n)}\) 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 \({U_q({\dot{{\mathfrak {gl}}}}_n)}\curvearrowright {{\mathbb {C}}}^n(z)\) and its usual R-matrix \(R(z) \in \text {End}({{\mathbb {C}}}^n \otimes {{\mathbb {C}}}^n)(z)\) (see Faddeev et al. in Leningrad Math J 1:193–226, 1990). An upshot of this construction is a new topological coproduct on \({U_{q,{{\overline{q}}}}(\ddot{{\mathfrak {gl}}}_n)}\) which extends the Drinfeld–Jimbo coproduct on the horizontal subalgebra \({U_q({\dot{{\mathfrak {gl}}}}_n)}\subset {U_{q,{{\overline{q}}}}(\ddot{{\mathfrak {gl}}}_n)}\).