A tale of two shuffle algebras

Abstract

As a quantum affinization, the quantum toroidal algebra 𝑈𝑞,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 Leningrad Math 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).