A tale of two shuffle algebras

Abstract

AbstractAs a quantum affinization, the quantum toroidal algebra $${U_{q,{{\overline{q}}}}(\ddot{{\mathfrak {gl}}}_n)}$$ U q , 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 $${U_q({\dot{{\mathfrak {gl}}}}_n)}\curvearrowright {{\mathbb {C}}}^n(z)$$ U q ( gl ˙ n ) ↷ C n ( z ) and its usual R-matrix $$R(z) \in \text {End}({{\mathbb {C}}}^n \otimes {{\mathbb {C}}}^n)(z)$$ R ( z ) ∈ End ( C n ⊗ 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 $${U_{q,{{\overline{q}}}}(\ddot{{\mathfrak {gl}}}_n)}$$ U q , q ¯ ( gl ¨ n ) which extends the Drinfeld–Jimbo coproduct on the horizontal subalgebra $${U_q({\dot{{\mathfrak {gl}}}}_n)}\subset {U_{q,{{\overline{q}}}}(\ddot{{\mathfrak {gl}}}_n)}$$ U q ( gl ˙ n ) ⊂ U q , q ¯ ( gl ¨ n ) .