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Representation-finite triangular algebras form an open scheme

Stanisław Kasjan (2003)

Open Mathematics

Let V be a valuation ring in an algebraically closed field K with the residue field R. Assume that A is a V-order such that the R-algebra Ā obtained from A by reduction modulo the radical of V is triangular and representation-finite. Then the K-algebra KA ≅ A ⊗V is again triangular and representation-finite. It follows by the van den Dries’s test that triangular representation-finite algebras form an open scheme.

Representations of a class of positively based algebras

Shiyu Lin, Shilin Yang (2023)

Czechoslovak Mathematical Journal

We investigate the representation theory of the positively based algebra A m , d , which is a generalization of the noncommutative Green algebra of weak Hopf algebra corresponding to the generalized Taft algebra. It turns out that A m , d is of finite representative type if d ≤ 4 , of tame type if d = 5 , and of wild type if d ≥ 6 . In the case when d ≤ 4 , all indecomposable representations of A m , d are constructed. Furthermore, their right cell representations as well as left cell representations of A m , d are described.

Representations of étale Lie groupoids and modules over Hopf algebroids

Jure Kališnik (2011)

Czechoslovak Mathematical Journal

The classical Serre-Swan's theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially...

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.

Restricted Boolean group rings

Dinesh Udar, R.K. Sharma, J.B. Srivastava (2017)

Archivum Mathematicum

In this paper we study restricted Boolean rings and group rings. A ring R is 𝑟𝑒𝑠𝑡𝑟𝑖𝑐𝑡𝑒𝑑𝐵𝑜𝑜𝑙𝑒𝑎𝑛 if every proper homomorphic image of R is boolean. Our main aim is to characterize restricted Boolean group rings. A complete characterization of non-prime restricted Boolean group rings has been obtained. Also in case of prime group rings necessary conditions have been obtained for a group ring to be restricted Boolean. A counterexample is given to show that these conditions are not sufficient.

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