About UMP algebras and a special classification case

TL;DR

This paper characterizes UMP algebras, focusing on locally monomial and special multiserial cases, and classifies symmetric special biserial UMP algebras using combinatorial and graph-theoretic methods.

Contribution

It introduces algebraic combinatorics tools to characterize UMP algebras, especially those that are locally monomial and special multiserial, and classifies symmetric special biserial UMP algebras.

Findings

Characterization of locally monomial UMP algebras

Description of ramifications graph of symmetric special biserial algebras

Classification of symmetric special biserial UMP algebras via bound quivers and Brauer graphs

Abstract

The class of UMP algebras arises in several classification problems in the context of derived categories of finite-dimensional algebras. In this paper we define the class of UMP algebras and develop algebraic combinatorics tools in order to present a characterization of this class of algebras which are locally monomial (see Definition 4.1) and special multiserial algebras. Among other things, we describe the ramifications graph of symmetric special biserial algebras and we classify which of them are UMP algebras in terms of their bound quivers and their associated Brauer graphs.