Primitivity Testing in Free Group Algebras via Duality

TL;DR

This paper develops explicit algorithms for primitivity testing and module decomposition in free group algebras, leveraging a novel duality concept within free ideal rings to enhance computational methods.

Contribution

It introduces a duality framework in free ideal rings and provides algorithms for primitivity testing, module splitting, and submodule intersection in free group algebras.

Findings

Algorithms determine primitivity of elements in free group algebras.

Methods for checking free summands and free splittings of modules.

Efficient computation of submodule intersections in free modules.

Abstract

Let $K$ be a field and $F$ a free group. By a classical result of Cohn and Lewin, the free group algebra $K\left[F\right]$ is a free ideal ring (FIR): a ring over which the submodules of free modules are themselves free, and of a well-defined rank. Given a finitely generated right ideal $I\leq K\left[F\right]$ and an element $f\in I$, we give an explicit algorithm determining whether $f$ is part of some basis of $I$. More generally, given free $K[F]$-modules $M\le N$, we provide algorithms determining whether $M$ is a free summand of $N$, and whether $N$ admits a free splitting relative to $M$. These can also be used to obtain analogous algorithms for free groups $H\le J$. As an aside, we also provide an algorithm to compute the intersection of two given submodules of a free $K\left[F\right]$-module. A key feature of this work is the introduction of a duality, induced by a matrix with entries in a free ideal ring, between the respective algebraic extensions of its column and row spaces.