YANGIANS, QUANTUM LOOP ALGEBRAS, AND ABELIAN DIFFERENCE EQUATIONS

• 成果类型: Article
• 署名作者: Gautam, Sachin; Laredo, Valerio Toledano
• 署名单位: Columbia University; Northeastern University
• 刊物名称: JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY
• ISSN/ISSBN: 0894-0347
• DOI: 10.1090/jams/851
• 发表日期: 2016
• 页码: 775-824
• 关键词: finite-dimensional representations; affine algebras; quiver varieties; SPECTRA; systems

摘要

Let g be a complex, semisimple Lie algebra, and Y-h(g) and U-q(Lg) the Yangian and quantum loop algebra of g. Assuming that h is not a rational number and that q = e(pi ih), we construct an equivalence between the finite-dimensional representations of U-q(Lg) and an explicit subcategory of those of Y-h(g) defined by choosing a branch of the logarithm. This equivalence is governed by the monodromy of the abelian, additive difference equations defined by the commuting fields of Y-h(g). Our results are compatible with q-characters, and apply more generally to a symmetrizable Kac-Moody algebra g, in particular to affine Yangians and quantum toroidal algebras. In this generality, they yield an equivalence between the representations of Y-h(g) and U-q(Lg) whose restriction to g and U(q)g, respectively, are integrable and in category O.