Categorical representation theory is filled with graded additive categories (defined by generators and relations) whose Grothendieck groups are algebras over $\mathbb{Z}[q,q^{-1}]$. For example, Khovanov-Lauda-Rouquier (KLR) algebras categorify the quantum group, and the diagrammatic Hecke categories categorify Hecke algebras. Khovanov introduced Hopfological algebra in 2006 as a method to potentially categorify the specialization of these $\mathbb{Z}[q,q^{-1}]-algebras$ at $q = \zeta_n$ a root of unity. The schtick is this: one equips the category (e.g. the KLR algebra) with a derivation d of degree 2, which satisfies d^p = 0 after specialization to characteristic p, making this specialization into a p-dg algebra. The p-dg Grothendieck group of a p-dg algebra is automatically a module over $\mathbb{Z}[\zeta_{2p}]$... but it is NOT automatically the specialization of the ordinary Grothendieck group at a root of unity!

Upgrading the categorification to a p-dg algebra was done for quantum groups by Qi-Khovanov and Qi-Elias. Recently, Qi-Elias accomplished the task for the diagrammatic Hecke algebra in type A, and ruled out the possibility for most other types. Now the question is: what IS the p-dg Grothendieck group? Do you get the quantum group/hecke algebra at a root of unity, or not?

This is a really hard question, and currently the only techniques for establishing such a result involve explicit knowledge of all the important idempotents in the category. These techniques sufficed for quantum sl_n with $n \le 3$, but new techniques are required to make further progress.

After reviewing the theory of p-dg algebras and their Grothendieck groups, we will present some new techniques and conjectures, which we hope will blow your mind.

Everything is joint with You Qi.

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