Representations of generalized bound path algebras

TL;DR

This paper extends the understanding of representations of generalized path algebras to include generalized bound path algebras, describing projective and injective modules within this broader framework.

Contribution

It introduces a description of representations of generalized bound path algebras, building on previous work to include relations and module types.

Findings

Representation descriptions for generalized bound path algebras

Characterization of projective modules

Characterization of injective modules

Abstract

The concept of generalized path algebras was introduced in (Coelho, Liu, 2000). Roughly speaking, these algebras are constructed in a similar way to that of the path algebras over a quiver, the difference being that we assign an algebra to each vertex of the quiver and consider paths intercalated with elements from these algebras. Then we use concatenation of paths together with the algebra structure in each vertex to define multiplication. The representations of a generalized path algebra were described in one of the main results of (Ib\'a\~nez Cobos et al., 2008), in terms of the representations of the algebras used in its construction. In this article, we continue our investigation started in (Chust, Coelho, 2021) and extend the result mentioned above to describe the representations of the generalized bound path algebras, which are a quotient of generalized path algebras by an ideal generated by relations. In particular, the representations associated with the projective and injective modules are described.