Derived equivalence classification of weakly symmetric algebras of Euclidean type.
The objective of the paper is a derived (and stable) equivalence classification of weakly symmetric algebras of Euclidean type. Let \(A\) be a basic finite dimensional algebra over an algebraically closed field \(K\). The algebra \(A\) is called weakly symmetric if the socle \(\text{soc\,}P\) of \(P\) is isomorphic to the top \(P/\text{rad\,}P\) for every indecomposable projective \(A\)-module \(P\). If \(A\) is symmetric, that is, \(A\) and \(\Hom_K(A,K)\) are isomorphic as \(A\)-\(A\)-bimodules, then it is weakly symmetric and every weakly symmetric algebra is selfinjective. A selfinjective algebra of Euclidean type is, by the definition due to \textit{A. Skowroński} [Math. Ann. 285, No. 2, 177-199 (1989; Zbl 0653.16021)], a quotient \(\widehat B/G\) of the repetitive algebra \(\widehat B\) of a tilted algebra \(B\) of Euclidean type by an action of an admissible group \(G\) of \(K\)-automorphisms of \(\widehat B\). A derived equivalence classification of weakly symmetric algebras of Euclidean type with singular Cartan matrix (they are derived/stably equivalent to trivial extensions of canonical algebras of Euclidean type) follows by results of \textit{I. Assem, J. Nehring} and \textit{A. Skowroński} [Tsukuba J. Math. 13, No. 1, 31-72 (1989; Zbl 0686.16011); Math. Ann. 280, No. 3, 441-463 (1988; Zbl 0617.16017)], \textit{L. Peng} and \textit{J. Xiao} [J. Lond. Math. Soc., II. Ser. 52, No. 1, 61-72 (1995; Zbl 0841.16019)] and \textit{A. Skowroński} and \textit{K. Yamagata} [Arch. Math. 70, No. 5, 341-350 (1998; Zbl 0915.16005)]. The main result of the present article asserts that a selfinjective algebra \(A\) of Euclidean type is weakly symmetric with nonsingular Cartan matrix if and only if \(A\) is derived/stably equivalent to one of the algebras \(A(\lambda)\), \(A(p,q)\), \(\Lambda(n)\), \(\Gamma(n)\), \(\lambda\in K^*\), \(1\leq p\leq q\), \(p+q\geq 3\), \(n\geq 2\), described explicitly in the paper.
scientific article; zbMATH DE number 2093438
scientific article