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Rank additivity for quasi-tilted algebras of canonical type

Thomas Hübner (1998)

Colloquium Mathematicae

Given the category X of coherent sheaves over a weighted projective line X = X ( λ , p ) (of any representation type), the endomorphism ring Σ = ( 𝒯 ) of an arbitrary tilting sheaf - which is by definition an almost concealed canonical algebra - is shown to satisfy a rank additivity property (Theorem 3.2). Moreover, this property extends to the representationinfinite quasi-tilted algebras of canonical type (Theorem 4.2). Finally, it is demonstrated that rank additivity does not generalize to the case of tilting complexes...

Rational smoothness of varieties of representations for quivers of Dynkin type

Philippe Caldero, Ralf Schiffler (2004)

Annales de l’institut Fourier

We study the Zariski closures of orbits of representations of quivers of type A , D ou E . With the help of Lusztig’s canonical base, we characterize the rationally smooth orbit closures and prove in particular that orbit closures are smooth if and only if they are rationally smooth.

Real representations of quivers

Lidia Hügeli, Sverre Smalø (1999)

Colloquium Mathematicae

The Dynkin and the extended Dynkin graphs are characterized by representations over the real numbers.

Recent results on quiver sheaves

Andreas Laudin, Alexander Schmitt (2012)

Open Mathematics

In this article, we survey recent work on the construction and geometry of representations of a quiver in the category of coherent sheaves on a projective algebraic manifold. We will also prove new results in the case of the quiver • ← • → •.

Representation-directed algebras form an open scheme

Stanislaw Kasjan (2002)

Colloquium Mathematicae

We apply van den Dries's test to the class of algebras (over algebraically closed fields) which are not representation-directed and prove that this class is axiomatizable by a positive quantifier-free formula. It follows that the representation-directed algebras form an open ℤ-scheme.

Representation-tame incidence algebras of finite posets

Zbigniew Leszczyński (2003)

Colloquium Mathematicae

Continuing the paper [Le], we give criteria for the incidence algebra of an arbitrary finite partially ordered set to be of tame representation type. This completes our result in [Le], concerning completely separating incidence algebras of posets.

Representation-tame locally hereditary algebras

Zbigniew Leszczyński (2004)

Colloquium Mathematicae

Let A be a finite-dimensional algebra over an algebraically closed field. The algebra A is called locally hereditary if any local left ideal of A is projective. We give criteria, in terms of the Tits quadratic form, for a locally hereditary algebra to be of tame representation type. Moreover, the description of all representation-tame locally hereditary algebras is completed.

Resolutions of homogeneous bundles on ℙ 2

Giorgio Ottaviani, Elena Rubei (2005)

Annales de l’institut Fourier

We characterize minimal free resolutions of homogeneous bundles on ℙ 2 . Besides we study stability and simplicity of homogeneous bundles on ℙ 2 by means of their minimal free resolutions; in particular we give a criterion to see when a homogeneous bundle is simple by means of its minimal resolution in the case the first bundle of the resolution is irreducible.

Ringel-Hall algebras of hereditary pure semisimple coalgebras

Justyna Kosakowska (2009)

Colloquium Mathematicae

We define and investigate Ringel-Hall algebras of coalgebras (usually infinite-dimensional). We extend Ringel's results [Banach Center Publ. 26 (1990) and Adv. Math. 84 (1990)] from finite-dimensional algebras to infinite-dimensional coalgebras.

Roots of Nakayama and Auslander-Reiten translations

Helmut Lenzing, Andrzej Skowroński (2000)

Colloquium Mathematicae

We discuss the roots of the Nakayama and Auslander-Reiten translations in the derived category of coherent sheaves over a weighted projective line. As an application we derive some new results on the structure of selfinjective algebras of canonical type.

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