Let K be a field of characteristic 0 containing all roots of unity. We classified all the Hopf structures on monomial K-coalgebras, or, in dual version, on monomial K-algebras.  2004 Elsevier Inc. All rights reserved.

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We associate to a localizable module a left retraction of algebras; it is a homological ring epimorphism that preserves singularity categories. We study the behavior of left retractions with respect to Gorenstein homological properties (for example, being Gorenstein algebras or CM-free).We apply the results to Nakayama algebras. It turns out that for a connected Nakayama algebra A, there exists a connected self-injective Nakayama algebra A ′ such that there is a sequence of left retractions linking A to A ′ ; in particular, the singularity category of A is triangle equivalent to the stable category of A ′ . We classify connected Nakayama algebras with at most three simple modules according to Gorenstein homological properties.

Abstract. We study finite quasi-quantum groups in their quiver setting developed recently by the first author. We obtain a classification of finite-dimensional pointed Majid algebras of finite corepresentation type, or equivalently a classification of elementary quasi-Hopf algebras of finite representation type, over the field of complex numbers. By the Tannaka-Krein duality principle, this provides a classification of the finite tensor categories in which every simple object has Frobenius-Perron dimension 1 and there are finitely many indecomposable objects up to isomorphism. Some interesting information of these finite tensor categories is given by making use of the quiver representation theory.