Some remarks about the ``double extension'' algebra of a finite poset. (Q2782436)

Let \(K\) be an algebraically closed field and \(S \equiv (S, \preceq)\) a finite partially ordered set. Denote by \(R=KS\) the incidence \(K\)-algebra of the poset \(S\). By a double extension \(K\)-algebra the author means the incidence \(K\)-algebra \(\widehat R = K \widehat S\) of the poset \( \widehat S = S \cup \{ \omega _ -, \omega _+\}\) obtained from \(S\) by adding the unique minimal element \(\omega _ -\) and the unique maximal element \( \omega _+\). The main result of the paper asserts that the double extension \(K\)-algebra \(\widehat R = K \widehat S\) is tilted if and only if the incidence \(K\)-algebra \( R = KS\) of \(S\) is tilted and there exists a tilting \(R\)-module \(T\) such that \(\text{End}_RT \) is hereditary and the indecomposable sincere \(R\)-module \(H\) with \(K\) at each vertex \(s\in S\) is a direct summand of \(T\).NEWLINENEWLINEFor the entire collection see [Zbl 0974.00038].