Cherednik Algebras and Affine Lie Algebras

表象理论诞生于大约120年前，融合了代数、分析、几何和数论的思想，其动机来自当时的当代物理学。它至今仍在数学中居于中心地位，并在生物、化学、信息技术，特别是在理论物理中得到了大量的应用。例如，正是表象理论为P.Higgs(2013年诺贝尔奖)在1964年做出的著名预测提供了数学基础，并在2012年得到了欧洲核子研究中心的实验人员的证实。反过来，物理学也仍然是表象理论的主要灵感来源。这与本项目特别相关，该项目旨在研究从可积系统理论和量子场论中产生的某些代数对象。这些代数对象是由I.Cherednik在1992年左右发现的，然后他用它们证明了I.Macdonald关于多元正交多项式的著名猜想。从那时起，这些代数就成为许多世界领先的数学家关注的焦点，并在代数几何、组合数学和数学物理中得到了惊人的应用。我们的目的是利用仿射李代数理论中的方法来推进这类代数的理论。

Representation Theory was born about 120 years ago by blending ideas from Algebra, Analysis, Geometry and Number Theory, with motivations coming from the then contemporary Physics. It keeps a central position in Mathematics to this day, and has found numerous applications in Biology, Chemistry, Information Technology and especially in Theoretical Physics. For instance, it was Representation Theory that gave mathematical foundations to the celebrated prediction made in 1964 by P. Higgs (Nobel Prize 2013) and confirmed by CERN experimenters in 2012. Reciprocally, Physics also remains a major source of inspiration for Representation Theory. This is particularly relevant to the present project, which is aimed to study certain algebraic objects emerged from the Theory of Integrable Systems and from the Quantum Field Theory. These algebraic objects were discovered around 1992 by I. Cherednik, who then used them to prove the famous conjectures of I. Macdonald on multivariate orthogonal polynomials. Since then these algebras have become the focus of attention of many world leading mathematicians, and found striking applications in Algebraic Geometry, Combinatorics and Mathematical Physics. We aim to advance the theory of these algebras by using the methods coming from the theory of affine Lie algebras.