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

本提案的主要主题是发现、解释和利用有限群表示理论与李论之间的联系。这些联系出现在许多不同的层次上：在Grothendieck群水平上的数值不变量之间的组合巧合，在梯度Grothendieck群水平上这些数值不变量的量化，以及在分类水平上的导出或稳定等价的Morita。在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等方面也有许多应用。提案中详细介绍了八个具体项目，其中许多项目通过分类的概念联系在一起。提案中的几个项目与有限w代数有关，这些代数在过去几年中作为重要对象出现在数学和数学物理的不同分支中。还有一些项目研究不同背景下的块和它们之间的等价关系，包括Broue猜想。最后讨论了在Aschbacher最大子群规划中的应用。广义地说，有限群论和李论是研究复杂结构对称性的理论。表征理论的思想是通过研究这些对称在已经被理解的更简单结构中的阴影来理解这些复杂的对称性。本提案中问题的动机之一是解释观察到的各种完全不同对象的表示理论之间的数值一致性，例如有限群代数块和李代数在潜在类别之间的等价。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。也有一些应用到数学和数学物理的其他领域，通过对象，如yangian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.