Continuous Nakayama Representations

TL;DR

This paper develops continuous analogues of Nakayama algebras using (pre-)Kupisch functions, establishing categorical equivalences under homeomorphisms and embedding finite-dimensional representations into continuous frameworks.

Contribution

It introduces continuous Nakayama representations and (pre-)Kupisch functions, and explores their categorical properties and connections to finite-dimensional Nakayama algebras.

Findings

Orientation-preserving homeomorphisms induce category equivalences.

Connectedness characterized by separation points.

Finite-dimensional representations embed into continuous categories.

Abstract

We introduce continuous analogues of Nakayama algebras. In particular, we introduce the notion of (pre-)Kupisch functions, which play a role as Kupisch series of Nakayama algebras, and view continuous Nakayama representations as a special type of representation of $\mathbb{R}$ or $\mathbb{S}^1$. We investigate equivalences and connectedness of the categories of Nakayama representations. Specifically, we prove that orientation-preserving homeomorphisms on $\mathbb{R}$ and on $\mathbb{S}^1$ induce equivalences between these categories. Connectedness is characterized by a special type of points called separation points determined by (pre-)Kupisch functions. We also construct an exact embedding from the category of finite-dimensional representations for any finite-dimensional Nakayama algebra, to a category of continuous Nakayama representaitons.