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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Yangians, quantum loop algebras, and abelian difference equations
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by Sachin Gautam and Valerio Toledano Laredo PDF
J. Amer. Math. Soc. 29 (2016), 775-824 Request permission

Abstract:

Let $\mathfrak {g}$ be a complex, semisimple Lie algebra, and $Y_\hbar (\mathfrak {g})$ and $U_q(L\mathfrak {g})$ the Yangian and quantum loop algebra of $\mathfrak {g}$. Assuming that $\hbar$ is not a rational number and that $q= e^{\pi i\hbar }$, we construct an equivalence between the finite-dimensional representations of $U_q(L\mathfrak {g})$ and an explicit subcategory of those of $Y_\hbar (\mathfrak {g})$ defined by choosing a branch of the logarithm. This equivalence is governed by the monodromy of the abelian, additive difference equations defined by the commuting fields of $Y_\hbar (\mathfrak {g})$. Our results are compatible with $q$-characters, and apply more generally to a symmetrizable Kac-Moody algebra $\mathfrak {g}$, in particular to affine Yangians and quantum toroïdal algebras. In this generality, they yield an equivalence between the representations of $Y_\hbar (\mathfrak {g})$ and $U_q(L\mathfrak {g})$ whose restriction to $\mathfrak {g}$ and $U_q\mathfrak {g}$, respectively, are integrable and in category $\mathcal {O}$.
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Additional Information
  • Sachin Gautam
  • Affiliation: Department of Mathematics, Columbia University, 2990 Broadway, New York, New York 10027
  • MR Author ID: 848064
  • Email: sachin@math.columbia.edu
  • Valerio Toledano Laredo
  • Affiliation: Department of Mathematics, Northeastern University, 567 Lake Hall, 360 Huntington Avenue, Boston, Massachusetts 02115
  • MR Author ID: 353547
  • Email: V.ToledanoLaredo@neu.edu
  • Received by editor(s): May 14, 2015
  • Published electronically: December 24, 2015
  • Additional Notes: The second author was supported in part by NSF Grants DMS-0854792, DMS-1206305, and PHY-1066293

  • Dedicated: To Andrei Zelevinsky (1953–2013). Advisor, mentor, colleague, and friend.
  • © Copyright 2015 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 29 (2016), 775-824
  • MSC (2010): Primary 17B37; Secondary 17B67, 39A10, 82B43
  • DOI: https://doi.org/10.1090/jams/851
  • MathSciNet review: 3486172