Large Non-Semisimple Categories in Representation Theory

这是表象理论的一个项目，它研究(物理)系统的对称性。表征理论最简单的表现形式是量纲分析，比如“把距离L和时间T组合成一个具有维度速度的量的方法就是取L/T；在这个例子中，重新缩放度量单位就成了一种对称。”一般而言，分析物理系统的对称性会导致对系统的不同组件如何交互进行非常细粒度的分析。经典的表示理论通过关注将系统分解成彼此之间不交互的组件(半简性)的问题，给出了简单而美丽的结果。另一种分解系统的方法是创建一个组件层次结构，“较大”组件影响“较小”组件，但在另一个方向上没有相互作用(就像在绕行星运行的质量可忽略的粒子中一样)。如今，已知许多对称类型的例子，对于这些对称所强迫的可能的层级，我们可以得到完整的描述。这个项目将开发新的方法来理解这些层次以及由此产生的相互作用。更详细地说，这个项目的主要目标是研究出现在李超代数、代数超群和无限维李代数的表示理论中的大范畴，重点是在抽象张量范畴理论中的应用。主要目标包括构造表示范畴的特定大界限，将它们描述为普适(在某种意义上)张量范畴，并从普遍性中提取这些表示范畴的详细描述。通过这些与大张量范畴、范畴和对偶的联系，这项工作旨在帮助解决关于超群表示的长期经典问题，例如特征标的计算、维度公式和Kazhdan-Lusztig理论。该项目的第二个目标是几何：将球形变元理论推广到超级情况。球面变元理论是代数几何和表示论的完美结合。由于缺乏半单性，为超群发展类似理论的问题是复杂的。最后，调和分析的问题，如分解函数空间和描述不变微分算符的谱，在现代理论物理中是必不可少的。该项目的第三个目标是开发一种通过超对称多项式和超几何理论来解决这些问题的方法。

This is a project in representation theory, which is the study of symmetries of (physical) systems. The simplest manifestation of representation theory is in dimensional analysis, as in "the way to combine a distance L and a time T into a quantity with dimension velocity is to take L/T;" in this example, rescaling the units of measure becomes a symmetry. In general, analyzing symmetries of physical systems leads to a very fine-grained analysis of how different components of a system may interact. Classical representation theory gives simple and beautiful results by focusing on the question of decomposing systems into components that do not interact between themselves (semisimplicity). Another way to decompose systems is to create a hierarchy of components, with "larger" components affecting "smaller" components, but no interaction in the other direction (as in a particle with negligible mass orbiting a planet). Nowadays many examples are known of types of symmetry for which we can obtain a complete description of possible hierarchies forced by these symmetries. This project will develop new ways to understand these hierarchies and the resulting interactions.In more detail, the main goal of this project is to study large categories appearing in representation theory of Lie superalgebras, algebraic supergroups, and infinite-dimensional Lie algebras, focusing on applications to the theory of abstract tensor categories. Key objectives include the construction of particular large limits of categories of representations, describing them as universal (in a certain sense) tensor categories, and extracting from universality a detailed description of these categories of representations. Via these connections with large tensor categories, categorification, and duality, this work aims to contribute to the solution of longstanding classical questions about representations of supergroups, such as the computation of characters, dimension formulae, and Kazhdan--Lusztig theory. A second goal of the project is geometric: to generalize the theory of spherical varieties to the super case. The theory of spherical varieties is a beautiful mix of algebraic geometry and representation theory. The question of developing an analogous theory for supergroups is complicated because of the lack of semisimplicity. Finally, questions of harmonic analysis, such as decomposing the space of functions and describing the spectra of invariant differential operators, are essential in modern theoretical physics. A third goal of the project is to develop an approach to these questions via theory of supersymmetric polynomials and supergeometry.