Green fields

TL;DR

This paper introduces Green fields, a special class of commutative Green biset functors with no non-trivial ideals, explores their properties, and provides conditions for semisimplicity and module category equivalences.

Contribution

It defines Green fields within Green biset functors, establishes criteria for their semisimplicity, and describes their module categories, advancing the understanding of their algebraic structure.

Findings

Green fields have no non-trivial ideals.

Criteria for semisimplicity of Green fields are established.

Certain Green fields have module categories equivalent to vector spaces over a field.

Abstract

We introduce {\em Green fields}, as commutative Green biset functors with no non-trivial ideals. We state some of their properties and give examples of known Green biset functors which are Green fields. Among the properties, we prove some criterions ensuring that a Green field is semisimple. Finally, we describe a type of Green field for which its category of modules is equivalent to a category of vector spaces over a field.