A tale of two shuffle algebras
Author(s)
NeguΘ, Andrei
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As a quantum affinization, the quantum toroidal algebra 𝑈𝑞,𝑞β―β―β―(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(𝔤𝔩Λ𝑛)β·β𝑛(𝑧)
and its usual R-matrix 𝑅(𝑧)∈End(β𝑛⊗β𝑛)(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,𝑞β―β―β―(𝔤𝔩¨𝑛)
which extends the Drinfeld–Jimbo coproduct on the horizontal subalgebra 𝑈𝑞(𝔤𝔩Λ𝑛)⊂𝑈𝑞,𝑞β―β―β―(𝔤𝔩¨𝑛).
Date issued
2024-06-25Department
Massachusetts Institute of Technology. Department of MathematicsPublisher
Springer Science and Business Media LLC
Citation
NeguΘ, A. A tale of two shuffle algebras. Sel. Math. New Ser. 30, 62 (2024).
Version: Final published version
ISSN
1022-1824
1420-9020