Flat Connections, Irregular Singularities and Quantum Groups

这一建议源于PI最近对kohnodrinfeld定理的推广，根据该定理，卡西米尔连接（在简单李代数的根超平面上具有对数奇点的平坦连接）的单一性由相应量子群的量子Weyl群算子描述。这个结果是由PI和De Concini独立推测出来的。第一个项目，与B. Feigin和E. Frenkel合作，通过展示Wess-Zumino-Witten模型中相关函数满足相应微分方程组的新场，努力建立卡西米尔连接和共形场论之间的精确联系，从而将具有不规则奇点的连接纳入该模型。除了保形场论之外，这个项目还有许多重要的潜在应用：简单李代数的结构和表示理论（参数子代数的位移的量化，不可约有限维表示的新基的构建）和统计力学（推广Gaudin模型的新可积模型的构建）。第二个项目旨在扩展π在单位根的单一性定理。这一扩展可以为广义编织张量范畴的研究提供动力，即其基础编织群是广义的张量范畴，其定义隐含在拟coxeter代数公理中。最后一个项目，与R. Rouquier合作，是关于Dynkin图上同调，这是由PI引入控制拟coxeter代数的变形。其目的是通过计算Coxeter群和半简单李代数的包络代数的上同调，达到对这种上同调更好的组合和拓扑理解。量子群是自然界最基本对称群的变形。它们是在八十年代中期作为某些量子力学系统的对称性被发现的，此后在数学和物理的广泛领域中出现，如弦理论、量子统计力学、有限对称群的研究、拓扑学和组合学。它们令人困惑的一个方面是，它们具有不可思议的能力，能够处理和解决这些领域中长期存在的问题，而这些问题通常是在没有量子群体的情况下制定的。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组解的分支行为。这一建议旨在进一步扩大我们对这一有趣现象的理解。该项目对弦理论和表示理论具有重要的潜在应用价值。它也有望导致编织张量范畴的广泛推广，以其目前的形式，已广泛应用于计算机科学，逻辑学以及最近的量子计算。PI曾多次向年轻的研究人员和研究生发表演讲，介绍他最近的研究成果，听众总是非常乐于接受。很有可能，上面所提到的由它们产生的进一步发展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.