# The center of a Green biset functor

## TL;DR

This paper introduces the concepts of commutant and center for Green biset functors, exploring their properties and applications in decomposing module categories, with explicit examples for classical shifted representation functors.

## Contribution

It defines the center and commutant of Green biset functors and demonstrates their use in category decomposition, extending ideas from Mackey functors.

## Key findings

- The center and commutant of a Green biset functor are well-defined and have useful properties.
- Category of modules over a Green biset functor can be decomposed into smaller categories.
- Explicit examples of such decompositions are provided for classical shifted representation functors.

## Abstract

For a Green biset functor $A$, we define the commutant and the center of $A$ and we study some of their properties and their relationship. This leads in particular to the main application of these constructions: the possibility of splitting the category of $A$-modules as a direct product of smaller abelian categories. We give explicit examples of such decompositions for some classical shifted representation functors. These constructions are inspired by similar ones for Mackey functors for a fixed finite group.