The representation-finite symmetric algebras with liftable simple modules
In [Ann. Math., II. Ser. 89, 209-241 (1969; Zbl 0197.02302)] \textit{G. J. Janusz} showed that a \(p\)-block algebra \(\Lambda\) of a finite group \(G\) of finite representation type over a splitting field \(F\) with characteristic \(p>0\) is uniserial if and only if every simple \(\Lambda\)-module \(M\) can uniquely be lifted to a simple \(\widehat{\Lambda}\otimes_RS\)-module, where \(R\) is a complete discrete rank one valuation ring with maximal ideal \(\pi R\), residue class field \(F=R/\pi R\), and quotient field \(S\) with characteristic zero, and where \(\widehat{\Lambda}\) is an \(RG\)-block such that \(\Lambda\cong\widehat\Lambda\otimes_RF\). This work is a generalization of this result to weakly symmetric finite dimensional algebras over an algebraically closed field, where the following is the main result in the paper. Let \(\Lambda\) be a connected, representation-finite, symmetric algebra over the algebraically closed field \(K\) with at least two non-isomorphic simple \(\Lambda\)-modules. Then the following conditions are equivalent: (1) \(\text{Ext}^2_\Lambda(M,M)=(0)\) for every simple \(\Lambda\)-module \(M\). (2) Either \(\Lambda\) is uniserial or \(c(\Lambda)=1\) and the hearts of all indecomposable projective \(\Lambda\)-modules are indecomposable. A key observation needed to prove this result is the following. For a weakly symmetric Artin algebra \(\Lambda\) if \(\text{Ext}^2_\Lambda(M,M)=(0)\) for a nonprojective simple module \(M\), then the heart of projective cover of \(M\) is indecomposable. The converse of this result is not true, but a partial converse is proven. In addition the nonuniserial algebras \(\Lambda\) with multiplicity \(c(\Lambda)=1\) satisfying condition (2) are classified.
weakly symmetric finite dimensional algebras
scientific article; zbMATH DE number 938558