On Higher Representation Theory via Categories of type Charge-Conserving--with--Glue

TL;DR

This paper develops a higher representation theory framework using categories of matrices, addressing non-semisimplicity and braid representations, with applications to classification and analysis of braid group representations.

Contribution

Introduces a strict monoidal subcategory of matrices for higher representation theory, extending classical concepts to non-semisimple contexts and braid representation analysis.

Findings

Established a new categorical framework for higher representation theory.

Applied the framework to classify braid representations.

Analyzed the structure of braid group representations in towers.

Abstract

In this paper we introduce a strict monoidal subcategory of the category of matrices, suitable to address a higher representation theoretic analogue of radicals (non-semisimplicity) in ordinary representation theory. We show the extent to which this analogue has analogous representation theoretic properties. To illustrate, we apply to two key problems in the study of braid representations (strict monoidal functors from the braid category $\mathsf{B}$ to the matrix category): the classification problem; and the problem of analysing the ordinary braid group representations that braid representations generate in towers.