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Journal of the American Mathematical Society

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Yangians, quantum loop algebras, and abelian difference equations


Authors: Sachin Gautam and Valerio Toledano Laredo
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
Published electronically: December 24, 2015
MathSciNet review: 3486172
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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.
Article copyright: © Copyright 2015 American Mathematical Society