On Simple Transitive 2-representations of Bimodules over the Dual Numbers

TL;DR

This paper classifies simple transitive 2-representations of bimodules over dual numbers, revealing their ranks and structures, and introduces new concepts for (co-)Duflo 1-morphisms in finitary 2-categories.

Contribution

It provides a classification of simple transitive 2-representations over dual numbers and proposes an alternative definition for (co-)Duflo 1-morphisms.

Findings

Rank 2 simple transitive 2-representations are exactly the cell 2-representations.

Rank 1 simple transitive 2-representations cannot be constructed via (co)algebra 1-morphisms.

Finitary apex representations are of rank 1 or 2.

Abstract

We study the problem of classification of simple transitive 2-representations for the (non-finitary) 2-category of bimodules over the dual numbers. We show that simple transitive 2-representations with finitary apex are necessarily of rank 1 or 2, and those of rank 2 are exactly the cell 2-representations. For 2-representations of rank 1, we show that they cannot be constructed using the approach of (co)algebra 1-morphisms. We also propose an alternative definition of (co-)Duflo 1-morphisms for finitary 2-categories and describe them in the case of bimodules over the dual numbers.