Prospecies of algebras I: Basic properties

TL;DR

This paper extends the theory of hereditary algebras to prospecies, a generalization of species, introducing new algebraic constructions and categorical equivalences.

Contribution

It develops the foundational properties of prospecies of algebras, including the construction of preprojective algebras and stable equivalences.

Findings

Constructed a preprojective algebra for prospecies

Established a stable equivalence between subcategories

Generalized hereditary algebra theory to prospecies

Abstract

In this paper, we generalise part of the theory of hereditary algebras to the context of prospecies of algebras. Here, a prospecies is a generalisation of Gabriel's concept of species gluing algebras via projective bimodules along a quiver to obtain a new algebra. This provides a categorical perspective on a recent paper by Gei\ss, Leclerc, and Schr\"oer. In particular, we construct a corresponding preprojective algebra, and establish a theory of a separated prospecies yielding a stable equivalence between certain functorially finite subcategories.