MIT Infinite Dimensional Algebra Seminar (Fall 2025)

Some additive monoidal categories admit an abelian envelope, which is a tensor category containing them in a universal way. Recently, a lot of theory has been developed to identify and construct abelian envelopes. This has been utilised to define the (higher) Verlinde categories ${\rm Ver}_{p^n}$, abelian envelopes of quotients of the category of tilting modules of ${\rm SL}_2$ in positive characteristic, and their union ${\rm Ver}_{p^\infty}$. The Verlinde categories are incompressible, meaning they cannot be expressed as the representations of an affine group scheme in any smaller category. In fact, the subcategories of ${\rm Ver}_{p^\infty}$ comprise all currently-known incompressible categories, making them a conjectural foundation for all symmetric tensor categories. I will give an overview of the construction of ${\rm Ver}_{p^\infty}$ by Benson-Etingof-Ostrik, and then discuss some adaptations to the construction which re-prove and elucidate some properties of these categories. This will give an alternate description of ${\rm Ver}_{p^\infty}$ in terms of the perfection of ${\rm SL}_2$, and an interpretation of ${\rm Ver}_{p^n}$ as a Serre quotient.

Semisimplification functors on tensor categories underlie many useful constructions in representation theory, including the Frobenius functor in modular representation theory and the Duflo-Serganova functor from Lie superalgebras. I will discuss how semisimplification functors and generalizations thereof can be applied to the study of modular representations and degenerate categorical Heisenberg actions. In the latter case, the functors will always define (non-exact!) morphisms of categorical actions, and in particular cases, categorify a certain element of the mod-p centre of affine sl_p. These functors admit a convenient diagrammatic description which clarifies their properties. Finally, I will explain how these functors naturally globalize known equivalences of subcategories of representations of S_n, originally due to Henke-Koenig and also studied by Harman.