A model for the canonical algebras of bimodules type (1, 4) over truncated polynomial rings

TL;DR

This paper characterizes simple regular representations of a species of type (1, 4) over complex Laurent series, providing normal forms and a model for associated canonical algebras using Galois descent techniques.

Contribution

It introduces a parametrization of simple regular representations of the species of type (1, 4) over complex Laurent series and proposes a model for the associated canonical algebras.

Findings

Parametrization of simple regular representations by closed points of a spectrum.

Provision of weak normal forms for these representations.

Description of canonical algebras associated to the species of type (1, 4).

Abstract

Let $k=\mathbb{C}(\!(\epsilon)\!)$ be the field of complex Laurent series. We use Galois descent techniques to show that the simple regular representations of the species of type $(1,\, 4)$ over $k$ are naturally parametrized by the closed points of $\mathrm{Spec}(k[x])\dot{\cup}\{1,\,2\}$. Moreover we provide weak normal forms for those representations. We use our representatives of the simple regular representations to describe the canonical algebras associated to the species of type (1, 4) over k. This suggest a model of those algebras in the sense of the work of Geiss, Leclerc and Schr\"oer [GLS17] and [GLS20].