Given a finite-dimensional algebra C (over an algebraically closed field) of global dimension at most two, we define its relation-extension algebra to be the trivial extension C ⋉ Ext C 2 ( D C , C ) of C by the C–C-bimodule Ext C 2 ( D C , C ) . We give a construction for the quiver of the relation-extension algebra in case the quiver of C has no oriented cycles. Our main result says that an algebra C ~ is cluster-tilted if and only if there exists a tilted algebra C such that C ~ is isomorphic to the relation-extension of C.
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Assem et al. (2008) studied this question.
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