Representations of finite groups and Lie algebras - from combinatorics to categorification

这个建议的主题是发现，解释和利用有限群的表示理论和李理论之间的联系。这些连接出现在许多不同的水平：组合的巧合之间的数值不变量的水平的Grothendieck集团，量化这些数值不变量的水平的分级Grothendieck集团，和森田，派生或稳定的等价在分类水平。也有许多应用，特别是有限群的结构和表示理论，有限维半单李代数，代数群和超群，量子群和无限维李代数。提案中详细列出了八个具体项目，其中许多项目通过分类的想法联系在一起。该提案中的几个项目涉及有限W-代数，这些代数在过去几年中作为数学和数学物理的各种分支中的重要对象而出现。也有项目研究块和它们之间的等价在各种情况下，包括布鲁猜想。最后有一个项目讨论Aschbacher的极大子群程序的应用。广义地说，有限群理论和李群理论都与研究复杂结构的对称性有关。表示论的思想是通过研究这些对称在已经被理解的简单结构中的影子来理解如此复杂的对称。在这个建议中的问题的动机之一是解释观察到的数字之间的巧合表示理论的各种完全不同的对象，如块有限群代数和李代数的基本范畴之间的等价关系。这一观点自然导致了范畴化的思想，其中组合不变量被范畴不变量所取代。也有一些应用到其他领域的数学和数学物理，通过对象，如杨吉亚和有限W-代数。

The main theme of this proposal is to discover, explain and exploit connections between representation theory of finite groups and Lie theory. These connections arise at many different levels: combinatorial coincidences between numerical invariants at the level of Grothendieck groups, quantizations of these numerical invariants at the level of graded Grothendieck groups, and Morita, derived or stable equivalences at the categorical level. There are also many applications, especially to the structure and representation theory of finite groups, finite dimensional semisimple Lie algebras, algebraic groups and supergroups, and to quantum groups and infinite dimensional Lie algebras. There are eight specific projects detailed in the proposal, many of which are linked together by the idea of categorification. Several projects in the proposal are concerned with finite W-algebras, which have emerged in the last few years as important objects in surprisingly diverse branches of mathematics and mathematical physics. There are also projects studying blocks and equivalences between them in various contexts, including Broue's conjecture. Finally there is one project discussing applications to Aschbacher's maximal subgroups program.Broadly speaking, finite group theory and Lie theory are concerned with studying the symmetry of complicated structures. The idea of representation theory is to understand such complicated symmetries by studying the shadows of these symmetries in simpler structures that are already understood. One of the motivations for the problems in this proposal is to explain observed numerical coincidences between the representation theory of various quite different objects such as blocks of finite group algebras and Lie algebras in terms of equivalences between the underlying categories. This point of view leads naturally to the idea of categorification in which combinatorial invariants are replaced by categorical ones. There are also a number of applications to other areas of mathematics and mathematical physics, via objects such as Yangians and finite W-algebras.