Flat Connections, Irregular Singularities and Quantum Groups

这个建议源于PI最近推广的Kohno-Drinfeld定理，根据该定理，Casimir连接的单值性，一个简单李代数根超平面上具有对数奇点的平坦连接，由相应量子群的量子Weyl群算子描述。该结果已由PI和De Concini独立证实。第一个项目，与B合作。Feigin和E. Frenkel，努力建立一个精确的联系之间的卡西米尔连接和共形场理论展示新领域的韦斯-祖米诺-维滕模型的相关函数满足相应的系统的微分方程，从而将连接与不规则奇点到这个模型。这个项目有一些重要的潜在应用以外的共形场论：结构和表示理论的简单李代数（量子化的转移的论点子代数，建设新的基础上不可约有限维表示）和统计力学（建设新的可积模型推广的高丁模型）。第二个项目的目的是在单位根上扩展PI的单值定理。这种扩展可以为广义辫子张量范畴的研究提供动力，这是张量范畴的基础辫子群是一个广义的，其定义是隐含在一个准考克斯特代数的公理。最后一个项目是与R. Rouquier，关注Dynkin图上同调，它是由PI引入的，用于控制拟Coxeter代数的变形。它的目的是通过计算Coxeter群和半单李代数的包络代数来达到对这种上同调的更好的组合和拓扑理解。量子群是自然界最基本的对称群的变形。它们是在80年代中期作为某些量子力学系统的对称性被发现的，此后出现在数学和物理学的广泛领域，如弦论，量子统计力学，有限对称群的研究，拓扑学和组合学。他们令人困惑的一个方面是他们的不可思议的能力，以解决这些领域的长期存在的问题，这些问题往往是制定没有呼吁量子群。PI一直在追求这样一种研究途径，通过使用量子群来描述复域中某些微分方程系统解的分支行为。这项建议旨在进一步扩大我们对这一有趣现象的理解。这个项目对弦论和表示论有重要的潜在应用。预计它还将导致编织张量范畴的广泛推广，这些范畴在目前的伪装下已广泛用于计算机科学，逻辑和最近的量子计算。PI曾多次向年轻研究人员和研究生的听众介绍他最近的研究结果，并总能找到一个非常容易接受的听众。这是可能的进一步发展所产生的，他们提出的上述将吸引这些年轻的数学家。

This proposal stems from the PI's recent generalisation of the Kohno-Drinfeld theorem according to which the monodromy of the Casimir connection, a flat connection with logarithmic singularities on the root hyperplanes of a simple Lie algebra, is described by the quantum Weyl group operators of the corresponding quantum group. This result had been conjectured independently by the PI and De Concini. The first project, in collaboration with B. Feigin and E. Frenkel, endeavours to establish a precise link between the Casimir connection and Conformal Field Theory by exhibiting new fields in the Wess-Zumino-Witten model whose correlation functions satisfy the corresponding system of differential equations, thus incorporating connections with irregular singularities into this model. This project has a number of important potential applications beyond those to Conformal Field Theory: to the structure and representation theory of simple Lie algebras (quantisation of the shift of argument subalgebra, construction of new basis of irreducible finite dimensional representations) and to Statistical Mechanics (construction of new integrable models generalising the Gaudin model). The second project aims at extending the PI's monodromy theorem at roots of unity. This extension could provide the impetus for the study of generalised braided tensor categories, that is tensor categories whose underlying braid group is a generalised one, and whose definition is implicit in the axioms of a quasi-Coxeter algebra. The last project, in collaboration with R. Rouquier, is concerned with Dynkin diagram cohomology, which was introduced by the PI to control deformations of quasi-Coxeter algebras. Its aim is to arrive at a better combinatorial and topological understanding of this cohomology by computing it for Coxeter groups and the enveloping algebras of semi-simple Lie algebras.Quantum groups are deformations of the most basic symmetry groups of Nature. They were discovered in the mid-eighties as symmetries of certain Quantum mechanical systems and have since appeared in a wide range of fields in Mathematics and Physics such as String Theory, Quantum Statistical Mechanics, the study of finite symmetry groups, Topology and Combinatorics. One of their bewildering aspects is their uncanny ability to address, and solve, long standing problems in these fields which are often formulated without appealling to quantum groups. The PI has been pursuing one such avenue of investigation, by using quantum groups to describe the branching behaviour of solutions of certain systems of differential equations in the complex domain. This proposal aims at further broadening our understanding of this intriguing phenomenon. This project has important potential applications to String Theory and Representation Theory. It is also expected that it will lead to a wide generalisation of Braided Tensor Categories which, in their present guise, have been extensively used in Computer Science, Logic and, more recently, Quantum Computing. The PI has, on several occasions, given lectures to audiences of young researchers and graduate students about his recent results and always found an extremely receptive audience. It is likely that the further developments stemming from them which are presented above will attract such young mathematicians.