An application of a theorem of Sheila Brenner for Hochschild extension algebras of a truncated quiver algebra (Q2071270)

In this paper, for a truncated quiver algebra \(A\) such that any oriented cycle is zero in \(A\), the author gives a similar interpretation of the numbers considered by \textit{S. Brenner} [Arch. Math. 62, No. 3, 203--206 (1994; Zbl 0796.16016)] for a Hochschild extension algebra. First, the author reviews the basic definitions and relevant facts about Hochschild extension algebras. He also recalls the results of Brenner. Then, he gives a characterization of nonzero oriented cycles in a Hochschild extension algebra. Finally, the author gives the number of cycles by computing the cardinality of the set of equivalence classes.