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Yangians, quantum loop algebras and elliptic quantum groups

Banff International Research Station for Mathematical Innovation and Discovery

The Yangian Yg and quantum loop algebra Uq(Lg) of a complex semisimple Lie algebra g share very many similarities, and were long thought to have the same representations, though no precise relation between them existed until recently. I will explain how to construct a faithful functor from the finite-dimensional representations of Yg to those of Uq(Lg). The functor is entirely explicit, and governed by the monodromy of the abelian difference equations determined by the commuting fields of the Yangian. It yields a meromorphic, braided Kazhdan-Lusztig equivalence between finite-dimensional representations of the Yg and of U_q(Lg). A similar construction yields a faithful functor from representations of U_q(Lg) to those of the elliptic quantum group E_{q,t}(g) corresponding to g. This allows in particular a classification of irreducible finite-dimensional representations of E_{q,tau}(g), which was previously unknown. This is joint work with Sachin Gautam (Perimeter Institute).

Mathematics; Nonassociative rings and algebras; Associative rings and algebras; Representation theory

video/mp4

Author affiliation: Northeastern University

2016-08-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/58754

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/