Reflexive serial algebras
Let R be a finite-dimensional algebra over a field K and M a left R- module of K-dimension n. Let \(\phi =\phi_ M: R\to M_ n(K)\) be the matrix representation of R afforded by M and A the subalgebra of \(M_ n(K)\) consisting of all linear transformations of M which leave all R- submodules of the K-vector space M invariant. Then \(\phi\) (R)\(\subset A\). If \(\phi (R)=A\), M is called reflexive. The paper is concerned with faithful reflexive R-modules in case R is a Nakayama \((=generalized\) uniserial) algebra and K is algebraically closed. Main result: Let L be the minimal faithful R-module, \(n=number\) of simple R-modules. (i) If L is indecomposable, then \(L\oplus (rad L)\) n is reflexive and is a subquotient module of every reflexive R-module. (ii) If L is not indecomposable, then every faithful R-module is reflexive.
scientific article; zbMATH DE number 4040113