Large Non-Semisimple Categories in Representation Theory

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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.