Homological embeddings for preprojective algebras

TL;DR

This paper classifies homological and weakly homological representation embeddings for preprojective algebras of Dynkin type, and explores conditions under which such classifications are possible for self-injective algebras.

Contribution

It provides an explicit classification of all weakly homological and homological embeddings for preprojective algebras of Dynkin type and links the classification to the Tachikawa conjecture for self-injective algebras.

Findings

Explicit classification of embeddings for Dynkin preprojective algebras

Connection between Tachikawa conjecture and classification for self-injective algebras

Identification of conditions for homological embeddings to exist

Abstract

For a fixed finite dimensional algebra $A$, we study representation embeddings of the form $mod(B)\rightarrow mod(A)$. Such an embedding is called homological, if it induces an isomorphism on all Ext-groups and weakly homological, if only Ext$^1$ is preserved. In case $A$ is a preprojective algebra of Dynkin type, we give an explicit classification of all weakly homological and homological embeddings. Furthermore, we show that for self-injective algebras a classification of homological embeddings becomes accessible once these algebras fulfil the Tachikawa conjecture.