On the first Hochschild cohomology of admissible algebras

TL;DR

This paper explores the first Hochschild cohomology of admissible algebras, generalizing basic algebras, by analyzing differential operators and providing explicit bases for certain classes of algebras.

Contribution

It characterizes the first Hochschild cohomology of admissible algebras, including a dimension formula and explicit bases for specific cases.

Findings

Dimension formula for first Hochschild cohomology

Explicit bases for acyclic complete monomial algebras

Explicit bases for acyclic truncated quiver algebras

Abstract

Our aim in this paper is to investigate the first Hochschild cohomology of {\em admissible algebras} which can be seen as a generalization of basic algebras. For this purpose, we study differential operators on an admissible algebra. Firstly, differential operators from a path algebra to its quotient algebra as an admissible algebra are discussed. Based on this discussion, the first cohomology with admissible algebras as coefficient modules is characterized, including their dimension formula. Besides, for planar quivers, the $k$-linear bases of the first cohomology of acyclic complete monomial algebras and acyclic truncated quiver algebras are constructed over the field $k$ of characteristic $0$.