# Maximal Subalgebras of Finite-Dimensional Algebras

## TL;DR

This paper classifies maximal subalgebras of finite-dimensional algebras over fields, providing a comprehensive framework that includes semisimple and non-semisimple cases, with applications to representation theory.

## Contribution

It offers a complete classification of maximal subalgebras, especially in the semisimple case, and extends results to non-semisimple algebras, including those presented by quivers with relations.

## Key findings

- Complete classification in the semisimple case
- Lifting classification to non-semisimple algebras
- Relations between algebra properties and subalgebras

## Abstract

We study maximal subalgebras of an arbitrary finite dimensional algebra over a field, and obtain full classification/description results of such algebras. This is done by first obtaining a complete classification in the semisimple case, and then lifting to non-semisimple algebras. The results are sharpest in the case of algebraically closed fields, and take special forms for algebras presented by quivers with relations. We also relate representation theoretic properties of the algebra and its maximal and other subalgebras, and provide a series of embeddings between quivers, incidence algebras and other structures which relate indecomposable representations of algebras and some subalgebras via induction/restriction functors. Some results in literature are also re-derived as a particular case, and other applications are given.