# The periplectic Brauer algebra III: The Deligne category

## TL;DR

This paper constructs a categorical representation of an infinite Temperley-Lieb algebra within a periplectic Deligne category, classifies tensor ideals, and analyzes indecomposable summands related to the periplectic Lie supergroup.

## Contribution

It introduces a faithful categorical representation of an infinite Temperley-Lieb algebra on the periplectic Deligne category and classifies indecomposable summands in tensor powers.

## Key findings

- Classified thick tensor ideals in the periplectic Deligne category
- Determined objects in the kernel of the monoidal functor to the module category
- Identified projective indecomposable summands and their simple tops

## Abstract

We construct a faithful categorical representation of an infinite Temperley-Lieb algebra on the periplectic analogue of Deligne's category. We use the corresponding combinatorics to classify thick tensor ideals in this periplectic Deligne category. This allows to determine the objects in the kernel of the monoidal functor going to the module category of the periplectic Lie supergroup. We use this to classify indecomposable direct summands in the tensor powers of the natural representation, determine which are projective and determine their simple top.