On the First Hochschild Cohomology linebreak of Admissible Algebras

The aim of this paper is to investigate the first Hochschildcohomology of admissible algebras which can be regarded as ageneralization of basic algebras. For this purpose, the authorsstudy differential operators on an admissible algebra. Firstly,differential operators from a path algebra to its quotient algebraas an admissible algebra are discussed. Based on this discussion,the first cohomology with admissible algebras as coefficient modulesis characterized, including their dimension formula. Besides, forplanar quivers, the $k$-linear bases of the first cohomology ofacyclic complete monomial algebras and acyclic truncated quiveralgebras are constructed over the field $k$ of characteristic $0$.