The first Hochschild cohomology group of a cluster-tilted algebra revisited

TL;DR

This paper investigates the structure of the first Hochschild cohomology group of cluster-tilted algebras, providing new homological insights and decompositions related to their tilted algebra components.

Contribution

It establishes an isomorphism for HH1(B) in terms of HH1(C) and HH1(B,E), offering a novel homological interpretation for these algebraic structures.

Findings

HH1(B) is isomorphic to HH1(C) plus HH1(B,E)

Provides homological interpretation for previous results

Enhances understanding of cluster-tilted algebra cohomology

Abstract

Given a cluster-tilted algebra B we study its first Hochschild cohomology group HH1(B) with coefficients in the B-B-bimodule B. If C is a tilted algebra such that B is the relation extension of C by E= Ext2(DC,C), then we prove that HH1(B) is isomorphic, as a vector space, to the direct sum of HH1(C) with HH1(B,E). This yields homological interpretations for results of the first and the fourth author with M.J. Redondo.