Interpolations of monoidal categories and algebraic structures by invariant theory

TL;DR

This paper develops a framework for interpolating algebraic structures and symmetric monoidal categories using invariant theory, generalizing known categories and connecting to TQFT constructions.

Contribution

It introduces a method to realize algebraic structures as characters of a Hopf algebra in symmetric monoidal categories, extending Deligne's categories and relating to TQFTs.

Findings

Characterization of structures via characters in symmetric monoidal categories

Construction of categories $C_{\chi}$ analogous to TQFT universal constructions

Identification of conditions for characters to originate from abelian categories

Abstract

In a previous work by the author it was shown that every finite dimensional algebraic structure over an algebraically closed field of characteristic zero K gives rise to a character $K[X]_{aug}\to K$, where $K[X]_aug$ is a commutative Hopf algebra that encodes scalar invariants of structures. This enabled us to think of some characters $K[X]_{aug}\to K$ as algebraic structures with closed orbit. In this paper we study structures in general symmetric monoidal categories, and not only in $Vec_K$. We show that every character $\chi : K[X]_{aug}\to K$ arises from such a structure, by constructing a category $C_{\chi}$ that is analogous to the universal construction from TQFT. We then give necessary and sufficient conditions for a given character to arise from a structure in an abelian category with finite dimensional hom-spaces. We call such characters good characters. We show that if $\chi$ is good then $C_{\chi}$ is abelian and semisimple, and that the set of good characters forms a K-algebra. This gives us a way to interpolate algebraic structures, and also symmetric monoidal categories, in a way that generalizes Deligne's categories $Rep(S_t)$, $Rep(GL_t(K))$, $Rep(O_t)$, and also some of the symmetric monoidal categories introduced by Knop. We also explain how one can recover the recent construction of 2 dimensional TQFT of Khovanov, Ostrik, and Kononov, by the methods presented here. We give new examples, of interpolations of the categories $Rep(Aut_{O}(M))$ where $O$ is a discrete valuation ring with a finite residue field, and M is a finite module over it. We also generalize the construction of wreath products with $S_t$, which was introduced by Knop.