Meromorphic tensor equivalence for Yangians and quantum loop algebras

Gautam, Sachin; Toledano Laredo, Valerio

doi:10.1007/s10240-017-0089-9

Gautam, Sachin ¹ ; Toledano Laredo, Valerio 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, Tome 125 (2017), pp. 267-337.

Résumé

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 $Y_{ħ} (g)$ .

MR Zbl

DOI : 10.1007/s10240-017-0089-9

Affiliations des auteurs :

Gautam, Sachin ¹ ; Toledano Laredo, Valerio 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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     title = {Meromorphic tensor equivalence for {Yangians} and quantum loop algebras},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {267--337},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {125},
     year = {2017},
     doi = {10.1007/s10240-017-0089-9},
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     mrnumber = {3668651},
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     url = {http://www.numdam.org/articles/10.1007/s10240-017-0089-9/}
}

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Gautam, Sachin; Toledano Laredo, Valerio. Meromorphic tensor equivalence for Yangians and quantum loop algebras. Publications Mathématiques de l'IHÉS, Tome 125 (2017), pp. 267-337. doi : 10.1007/s10240-017-0089-9. http://www.numdam.org/articles/10.1007/s10240-017-0089-9/

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