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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
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

Valerio Toledano Laredo
Affiliation: Department of Mathematics, Northeastern University, 567 Lake Hall, 360 Huntington Avenue, Boston, Massachusetts 02115

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

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