A tale of two shuffle algebras

Abstract

As a quantum affinization, the quantum toroidal algebra Uq,q¯(gl¨n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${U_{q,{{\\overline{q}}}}(\\ddot{{\\mathfrak {gl}}}_n)}$$\\end{document} 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)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${U_q({\\dot{{\\mathfrak {gl}}}}_n)}\\curvearrowright {{\\mathbb {C}}}^n(z)$$\\end{document} and its usual R-matrix R(z)∈End(Cn⊗Cn)(z)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R(z) \\in \ ext {End}({{\\mathbb {C}}}^n \\otimes {{\\mathbb {C}}}^n)(z)$$\\end{document} (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)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${U_{q,{{\\overline{q}}}}(\\ddot{{\\mathfrak {gl}}}_n)}$$\\end{document} which extends the Drinfeld–Jimbo coproduct on the horizontal subalgebra Uq(gl˙n)⊂Uq,q¯(gl¨n)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${U_q({\\dot{{\\mathfrak {gl}}}}_n)}\\subset {U_{q,{{\\overline{q}}}}(\\ddot{{\\mathfrak {gl}}}_n)}$$\\end{document}.