On the existence of super-decomposable pure-injective modules over strongly simply connected algebras of non-polynomial growth. (Q2879356)

This paper continues and contributes to the extensive research on the relationship between the representation type of finite-dimensional algebras and their infinite-dimensional modules. A module \(M\) over a ring \(R\) is called super-decomposable if \(M\) has no indecomposable direct summands.NEWLINENEWLINE In [Ann. Pure Appl. Logic 26, 149-213 (1984; Zbl 0593.16019)], \textit{M. Ziegler} provided a fundamental criterion for a countable ring to have super-decomposable pure-injective modules. More recently, several authors studied the existence of super-decomposable pure-injective modules over finite-dimensional algebras over a field. For example, \textit{M. Prest} proved [in Model theory and modules. Cambridge: Cambridge University Press (1988; Zbl 0634.03025)] that super-decomposable pure-injective modules exist over strictly wild algebras. \textit{G. Puninski} [Proc. Am. Math. Soc. 132, No. 7, 1891-1898 (2004; Zbl 1133.16015)] showed that such modules exist over non-polynomial growth string algebras over a countable field. \textit{R. Harland} [Ph.D. Thesis, Univ. Manchester (2011)] proved that super-decomposable pure-injective modules exist over tubular algebras.NEWLINENEWLINE In the present paper, the authors apply their recent results [in Colloq. Math. 123, No. 2, 249-276 (2011; Zbl 1257.16013)] to show the existence of super-decomposable pure-injective modules for all non-polynomial growth strongly simply connected algebras over countable algebraically closed fields of characteristic different from \(2\). The proof of this result is based on the concept of independent pairs of dense chains of pointed modules [see \textit{G. Puninski, V. Puninskaya} and \textit{C. Toffalori}, J. Lond. Math. Soc., II. Ser. 78, No. 1, 125-142 (2008; Zbl 1155.20005)], and \textit{A. Skowroński}'s criterion of polynomial growth of strongly simply connected algebras [see Compos. Math. 109, No. 1, 99-133 (1997; Zbl 0889.16004)].