Constructing tilted algebras from cluster-tilted algebras.
It was proven by \textit{I. Assem, T. Brüstle} and \textit{R. Schiffler} [Bull. Lond. Math. Soc. 40, No. 1, 151-162 (2008; Zbl 1182.16009)] that any cluster-tilted algebra can be obtained from a tilted algebra as a certain semi-direct product called the ``relation extension'' of the tilted algebra. There is not a one-to-one correspondence between tilted algebras and cluster-tilted algebras; several tilted algebras may have isomorphic relation extensions. The article under review provides an algorithm to determine all tilted algebras whose relation extension is a given cluster-tilted algebra of finite type. There are several well-presented examples to illustrate the algorithm and the theory necessary to carry it out.
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