Meromorphic tensor equivalence for Yangians and quantum loop algebras
Publications Mathématiques de l'IHÉS, Tome 125 (2017), pp. 267-337.

Let g be a complex semisimple Lie algebra, and Yħ(g), Uq(Lg) the corresponding Yangian and quantum loop algebra, with deformation parameters related by q=eπιħ. 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 Yħ(g) to those of Uq(Lg). 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 Repfd(Yħ(g)) defined by choosing a branch of the logarithm. In this paper, we construct a tensor structure on Γ and show that, if |q|1, it yields an equivalence of meromorphic braided tensor categories, when Yħ(g) and Uq(Lg) 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 qKZ equations defined by Yħ(g). The tensor structure arises from the abelian qKZ equations defined by an appropriate regularisation of the commutative part of the R-matrix of Yħ(g).

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
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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://archive.numdam.org/articles/10.1007/s10240-017-0089-9/

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