Supercanonical algebras.
The authors introduce a new family of algebras known as supercanonical algebras. Up to derived equivalence, these algebras include the tame hereditary algebras, the tubular algebras, the canonical algebras, the concealed-canonical algebras and the quasitilted algebras of canonical type. Supercanonical algebras, while not necessarily possessing families of stable tubes which are separating or without external short cycles (i.e. certain nice properties), still have families of stable tubes with good properties. For more on canonical algebras, see \textit{W. Geigle} and \textit{H. Lenzing} [Lect. Notes Math. 1273, 265-297 (1987; Zbl 0651.14006)], \textit{C. M. Ringel} [Tame algebras and integral quadratic forms, (Lect. Notes Math. 1099) (1984; Zbl 0546.16013)]. The data required for the definition is an algebraically closed field \(k\), finite posets \(S_1,S_2,\dots,S_t\) and distinct non-zero elements \(\lambda_3,\lambda_4,\dots,\lambda_t\in k\) (\(t\geq 2\)). Let \(\widetilde S\) denote the double cone of any finite poset \(S\) (obtained by adding maximum and minimum elements to \(S\)). Form a new poset by identifying the maximum and minimum elements of all the double cones \(\widetilde{S_i}\). The supercanonical algebra is the corresponding quiver algebra subject to certain relations. The paper forms an extensive study of supercanonical algebras, including their relationship with known families of algebras, their Hochschild cohomology, the Grothendieck group, Euler form, Coxeter matrix, families of tubes, existence of preprojective components, and some of their derived properties.
scientific article; zbMATH DE number 2119566
scientific article