Galois covering technique and tame non-simply connected posets of polynomial growth
Let \(k\) be an algebraically closed field and let \(S\) be a finite partially ordered set (poset). Let \(Q_S\) be the quiver whose vertices are the elements of \(S\) and whose arrows are described as follows: for \(x,y\in S\) there is an arrow \(x\to y\) in \(Q_S\) if and only if \(x<y\) in \(S\). The set \(S\) is said to be simply connected if so is the factor algebra of the path algebra \(kQ_S\) modulo the ideal generated by all commutativity relations [\textit{I. Assem} and \textit{A. Skowroński}, Proc. Lond. Math. Soc., III. Ser. 56, No. 3, 417-450 (1988; Zbl 0617.16018)]. The authors show that if \(S\) is not simply connected, then it is of tame prinjective type and of polynomial growth over \(k\) if and only if: (a) the Tits form of \(S\) is weakly nonnegative and (b) no subset of \(S\) belongs to one of three specific families of posets. For the terminology, the reader is referred to \textit{D. Simson} [Linear representations of partially ordered sets and vector space categories. Algebra, Logic and Applications, 4. Gordon and Breach Science Publishers, Brooklyn (1992; Zbl 0818.16009)].
scientific article; zbMATH DE number 1421591