Meromorphic tensor equivalence for Yangians and quantum loop algebras

Sachin Gautam; Valerio Toledano Laredo

doi:10.1007/s10240-017-0089-9

Sachin Gautam ¹; Valerio Toledano Laredo ²

¹ Perimeter Institute for Theoretical Physics 31 Caroline Street N. N2L 2Y5 Waterloo ON Canada
² Department of Mathematics, Northeastern University 360 Huntington Avenue 02115 Boston MA USA

Publications Mathématiques de l'IHÉS, Volume 125 (2017), pp. 267-337

Abstract

Let $g$ be a complex semisimple Lie algebra, and $Y_{ħ} (g)$ , $U_{q} (L g)$ the corresponding Yangian and quantum loop algebra, with deformation parameters related by $q = e^{π ι ħ}$ . When $ħ$ is not a rational number, we constructed in Gautam and Toledano Laredo (J. Am. Math. Soc. 29:775, 2016) a faithful functor $Γ$ from the category of finite-dimensional representations of $Y_{ħ} (g)$ to those of $U_{q} (L g)$ . The functor $Γ$ is governed by the additive difference equations defined by the commuting fields of the Yangian, and restricts to an equivalence on a subcategory of ${Rep}_{fd} (Y_{ħ} (g))$ defined by choosing a branch of the logarithm. In this paper, we construct a tensor structure on $Γ$ and show that, if $| q | \neq 1$ , it yields an equivalence of meromorphic braided tensor categories, when $Y_{ħ} (g)$ and $U_{q} (L g)$ are endowed with the deformed Drinfeld coproducts and the commutative part of their universal $R$ -matrices. This proves in particular the Kohno–Drinfeld theorem for the abelian $q$ KZ equations defined by $Y_{ħ} (g)$ . The tensor structure arises from the abelian $q$ KZ equations defined by an appropriate regularisation of the commutative part of the $R$ -matrix of $Y_{ħ} (g)$ .

Received: 2015-10-16
Accepted: 2017-05-12
Online First: 2017-06-28
Published online: 2017-06-07

MR Zbl

DOI: 10.1007/s10240-017-0089-9

Author's affiliations:

Sachin Gautam ¹; Valerio Toledano Laredo ²

¹ Perimeter Institute for Theoretical Physics 31 Caroline Street N. N2L 2Y5 Waterloo ON Canada
² Department of Mathematics, Northeastern University 360 Huntington Avenue 02115 Boston MA USA

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     author = {Sachin Gautam and Valerio Toledano Laredo},
     title = {Meromorphic tensor equivalence for {Yangians} and quantum loop algebras},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {267--337},
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Sachin Gautam; Valerio Toledano Laredo. Meromorphic tensor equivalence for Yangians and quantum loop algebras. Publications Mathématiques de l'IHÉS, Volume 125 (2017), pp. 267-337. doi: 10.1007/s10240-017-0089-9

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