A tale of two shuffle algebras
Name
29_2024_Article_941.pdf
Size
3.41 MB
Format
Adobe PDF
Checksum (MD5)
38de464f79ebce7f142fc1575d30111c
Author(s)
Neguț, Andrei
Date Issued
June 25, 2024
Publisher
Springer Science and Business Media LLC
Citation
Neguț, A. A tale of two shuffle algebras. Sel. Math. New Ser. 30, 62 (2024).
Version
Final published version
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).
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.1007/s00029-024-00941-7
