Relative Adjoint Algebras

TL;DR

This paper introduces a method to construct exact commutative algebras in the center of a finite tensor category, using relative (co)ends to measure subcategory inclusions, with explicit examples.

Contribution

It develops a new approach to produce exact commutative algebras in the center of tensor categories via relative (co)ends, extending previous methods.

Findings

Explicit computations of the constructed algebras.

A new categorical tool for measuring subcategory inclusions.

Extension of Shimizu's construction using relative (co)ends.

Abstract

Given a finite tensor category $\ca$, an exact indecomposable $\ca$-module category $\Mo$, and a tensor subcategory $\Do\subseteq \ca^*_\Mo$, we describe a way to produce \textit{exact} commutative algebras in the center $Z(\ca)$, measuring this inclusion. The construction of such algebras is done in an analogous way as presented by Shimizu \cite{Sh2}, but using instead the \textit{relative (co)end}, a categorical tool developed in \cite{BM} in the realm of representations of tensor categories. We provide some explicit computations.