8 research outputs found
Meromorphic tensor equivalence for Yangians and quantum loop algebras
Let be a complex semisimple Lie algebra, and , the corresponding Yangian and quantum loop algebra,
with deformation parameters related by . When is not a
rational number, we constructed in arXiv:1310.7318 a faithful functor
from the category of finite-dimensional representations of to those of . The functor is governed by the
additive difference equations defined by the commuting fields of the Yangian,
and restricts to an equivalence on a subcategory of
defined by choosing a branch of the logarithm. In this paper, we construct a
tensor structure on and show that, if , it yields an
equivalence of meromorphic braided tensor categories, when
and are endowed with the deformed Drinfeld coproducts and
the commutative part of the universal -matrix. This proves in particular the
Kohno-Drinfeld theorem for the abelian KZ equations defined by
. The tensor structure arises from the abelian KZ
equations defined by a appropriate regularisation of the commutative -matrix
of .Comment: Title changed, details added. 67 pages, 1 figure. Final version, to
appear in Publ. Math IHE
THE -ANALOGUE OF THE WILD FUNDAMENTAL GROUP(I)(Algebraic, Analytic and Geometric Aspects of Complex Differential Equations and their Deformations. Painleve Hierarchies)
An introduction to difference Galois theory
International audienceThese are notes for my lectures at the summer school“Abecedarian of SIDE” held at CRM (Montr ́eal) in June 2016. Theyare intended to give a short introduction to difference Galois theory,leaving aside the technicalities
Functional relations of solutions of -difference equations
International audienceIn this paper, we study the algebraic relations satisfied by the solutions of -difference equations and their transforms with respect to an auxiliary operator. Our main tool is the parametrized Galois theories developed in two papers. The first part of this paper is concerned with the case where the auxiliary operator is a derivation, whereas the second part deals a -difference operator. In both cases, we give criteria to guaranty the algebraic independence of a series, solution of a -difference equation, with either its successive derivatives or its -transforms. We apply our results to -hypergeometric series