Twisted Gelfand-Ponomarev modules

TL;DR

This paper provides a self-contained classification of finite-dimensional vector spaces with two linear operators satisfying specific relations, extending previous work and applying it to $F$-crystals.

Contribution

It reworks the Gelfand-Ponomarev classification using Kraft quivers and offers an algebraic proof of a theorem on $F$-crystals.

Findings

Classification of $K$-vector spaces with operators $F$ and $V$ such that $FV=VF=0$

Reinterpretation of the classification via Kraft quivers

Algebraic proof of a theorem on the existence of $F$-crystals

Abstract

In this expository paper, given a field $K$ and two automorphisms $\sigma, \tau \in \mathrm{Aut}(K)$, we give a self-contained proof of the classification of finite dimensional $K$-vector spaces equipped with two operators $F$ and $V$, respectively $\sigma$-linear and $\tau$-linear, such that $FV = VF = 0$. This classification was originally due to the combined results of Gelfand and Ponomarev (1968), and of Kraft (1975). Following a recent suggestion of Chai (2025), we reworked their classification in light of the notion of Kraft quivers. As an application, we generalize and give an algebraic proof of a theorem by Kottwitz and Rapoport concerning the existence of $F$-crystals.