Representation-directed diamonds (Q2709397)

The author considers the local-colocal modules over a finite-dimensional algebra. These modules are called in the paper the diamond modules, here the main interest is to classify all diamonds for algebras of finite representation type. As pointed out by the author, this question can be reduced to that for representation-directed algebras by the covering theory developed by Bongartz and Gabriel. So the focus of the paper is on the representation-directed algebras with sincere diamonds. The main result of the paper is a criterion which says that a representation-directed algebra \(A\) over an algebraically closed field has a sincere diamond if and only if the vector \((1,1,\dots,1)\) is the only sincere positive \(1\)-root of the Tits form of \(A\). Based on results in this paper and some other known results, the author has a classification list of all representation-directed algebras with faithful diamonds.