# On the first Hochschild cohomology of admissible algebras

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## Abstract

The aim of this paper is to investigate the first Hochschild cohomology of admissible algebras which can be regarded as a generalization of basic algebras. For this purpose, the authors 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.

## Keywords

Quiver Admissible algebra Differential operators Cohomology## 2000 MR Subject Classification

16E40 16G20 16W25 17B70## Preview

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