Large Non-Semisimple Categories in Representation Theory

这是一个表示论的项目，它是对（物理）系统对称性的研究。表示论最简单的表现形式是量纲分析，如“将距离L和时间T联合收割机组合成具有量纲速度的量的方法是取L/T”;在这个例子中，重新标度度量单位成为一种对称。一般来说，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。经典表示论通过专注于将系统分解为彼此之间不相互作用的组件（半简单性）的问题，给出了简单而美丽的结果。分解系统的另一种方法是创建一个组件层次，“较大”的组件影响“较小”的组件，但在另一个方向上没有相互作用（如在一个质量可以忽略不计的粒子绕行星运行）。如今，我们已经知道了许多对称类型的例子，我们可以对由这些对称所导致的可能的等级结构进行完整的描述。本课题的主要目的是研究李超代数、代数超群、无限维李代数的表示论中出现的大范畴，并将其应用于抽象张量范畴的理论。主要目标包括构建表征范畴的特别大的极限，将它们描述为普适的（在某种意义上）张量范畴，并从普适性中提取这些表征范畴的详细描述。通过这些与大张量范畴，范畴化和对偶的联系，这项工作的目的是有助于解决关于超群表示的长期经典问题，例如特征的计算，维数公式和Kazhdan-Lusztig理论。该项目的第二个目标是几何：推广理论的球形品种的超情况。球面簇的理论是代数几何和表示论的完美结合。由于缺乏半简单性，发展超群的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子的谱，在现代理论物理中是必不可少的。该项目的第三个目标是通过超对称多项式和超几何理论来解决这些问题。

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.