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 ${U_q({\dot{{\mathfrak {gl}}}}_n)}\subset {U_{q,{{\overline{q}}}}(\ddot{{\mathfrak {gl}}}_n)}$.