On selfadjoint functors satisfying polynomial relations

TL;DR

This paper investigates selfadjoint functors on module categories over finite dimensional algebras that satisfy polynomial relations, classifying simple cases and exploring constructions for new actions.

Contribution

It classifies selfadjoint functors satisfying basic polynomial relations and introduces methods to construct new functor actions using algebraic operations.

Findings

Classification of idempotent and square root functors

Descriptions of natural constructions for new actions

Analysis of polynomial relations in functor categories

Abstract

We study selfadjoint functors acting on categories of finite dimensional modules over finite dimensional algebras with an emphasis on functors satisfying some polynomial relations. Selfadjoint functors satisfying several easy relations, in particular, idempotents and square roots of a sum of identity functors, are classified. We also describe various natural constructions for new actions using external direct sums, external tensor products, Serre subcategories, quotients and centralizer subalgebras.