The geometry of uniserial representations of finite dimensional algebras I

TL;DR

This paper explores the geometric structure of uniserial modules over finite dimensional algebras, providing a classification via algebraic varieties, algorithms for isomorphism problems, and new invariants, thereby deepening understanding of their representation theory.

Contribution

It introduces a finite list of algebraic varieties that classify uniserial modules and provides constructive methods and invariants for their analysis.

Findings

Finite varieties correspond to uniserial module types

An algorithm for resolving uniserial module isomorphism

Every affine variety can be realized as a uniserial module variety

Abstract

It is shown that, given any finite dimensional, split basic algebra $\Lambda = K\Gamma/I$ (where $\Gamma$ is a quiver and $I$ an admissible ideal in the path algebra $K \Gamma$), there is a finite list of affine algebraic varieties, the points of which correspond in a natural fashion to the isomorphism types of uniserial left $\Lambda$-modules, and the geometry of which faithfully reflects the constraints met in constructing such modules. A constructive coordinatized access to these varieties is given, as well as to the accompanying natural surjections from the varieties onto families of uniserial modules with fixed composition series. The fibres of these maps are explored, one of the results being a simple algorithm to resolve the isomorphism problem for uniserial modules. Moreover, new invariants measuring the complexity of the uniserial representation theory are derived from the geometric viewpoint. Finally, it is proved that each affine algebraic variety arises as a variety of uniserial modules over a suitable finite dimensional algebra, in a setting where the points are in one-one correspondence with the isomorphism classes of uniserial modules.