A bocs theoretic characterization of gendo-symmetric algebras

TL;DR

This paper characterizes gendo-symmetric algebras using bocs-structures, establishing a correspondence between algebra properties and bocs theory, and explores module category isomorphisms related to these algebras.

Contribution

It provides a bocs-theoretic characterization of gendo-symmetric algebras and links their module categories to those of certain subalgebras, offering new insights into their structure.

Findings

Gendo-symmetric algebras correspond to bocs-structures on (A,D(A)).

Module categories of these bocses are isomorphic to those of eAe for specific idempotents.

New results about gendo-symmetric algebras are derived using bocs theory.

Abstract

Gendo-symmetric algebras were recently introduced by Fang and K\"onig. An algebra is called gendo-symmetric in case it is isomorphic to the endomorphism ring of a generator over a finite dimensional symmetric algebra. We show that a finite dimensional algebra $A$ over a field $K$ is gendo-symmetric if and only if there is a bocs-structure on $(A,D(A))$, where $D=Hom_K(-,K)$ is the natural duality. Assuming that $A$ is gendo-symmetric, we show that the module category of the bocs $(A,D(A))$ is isomorphic to the module category of the algebra $eAe$, when $e$ is an idempotent such that $eA$ is the unique minimal faithful projective-injective right $A$-module. We also prove some new results about gendo-symmetric algebras using the theory of bocses.