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A categorification of the square root of -1

Yin Tian (2016)

Fundamenta Mathematicae

We give a graphical calculus for a monoidal DG category ℐ whose Grothendieck group is isomorphic to the ring ℤ[√(-1)]. We construct a categorical action of ℐ which lifts the action of ℤ[√(-1)] on ℤ².

A classification of two-peak sincere posets of finite prinjective type and their sincere prinjective representations

Justyna Kosakowska (2001)

Colloquium Mathematicae

Assume that K is an arbitrary field. Let (I, ⪯) be a two-peak poset of finite prinjective type and let KI be the incidence algebra of I. We study sincere posets I and sincere prinjective modules over KI. The complete set of all sincere two-peak posets of finite prinjective type is given in Theorem 3.1. Moreover, for each such poset I, a complete set of representatives of isomorphism classes of sincere indecomposable prinjective modules over KI is presented in Tables 8.1.

A computation of positive one-peak posets that are Tits-sincere

Marcin Gąsiorek, Daniel Simson (2012)

Colloquium Mathematicae

A complete list of positive Tits-sincere one-peak posets is provided by applying combinatorial algorithms and computer calculations using Maple and Python. The problem whether any square integer matrix A ( ) is ℤ-congruent to its transpose A t r is also discussed. An affirmative answer is given for the incidence matrices C I and the Tits matrices C ̂ I of positive one-peak posets I.

A family of noetherian rings with their finite length modules under control

Markus Schmidmeier (2002)

Czechoslovak Mathematical Journal

We investigate the category mod Λ of finite length modules over the ring Λ = A k Σ , where Σ is a V-ring, i.e. a ring for which every simple module is injective, k a subfield of its centre and A an elementary k -algebra. Each simple module E j gives rise to a quasiprogenerator P j = A E j . By a result of K. Fuller, P j induces a category equivalence from which we deduce that mod Λ j b a d h b o x P j . As a consequence we can (1) construct for each elementary k -algebra A over a finite field k a nonartinian noetherian ring Λ such that mod A mod Λ , (2) find twisted...

A Generalization of Baer's Lemma

Molly Dunkum (2009)

Czechoslovak Mathematical Journal

There is a classical result known as Baer’s Lemma that states that an R -module E is injective if it is injective for R . This means that if a map from a submodule of R , that is, from a left ideal L of R to E can always be extended to R , then a map to E from a submodule A of any R -module B can be extended to B ; in other words, E is injective. In this paper, we generalize this result to the category q ω consisting of the representations of an infinite line quiver. This generalization of Baer’s Lemma...

A remark on quiver varieties and Weyl groups

Andrea Maffei (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we define an action of the Weyl group on the quiver varieties M m , λ ( v ) with generic ( m , λ ) .

Abelian group pairs having a trivial coGalois group

Paul Hill (2008)

Czechoslovak Mathematical Journal

Torsion-free covers are considered for objects in the category q 2 . Objects in the category q 2 are just maps in R -Mod. For R = , we find necessary and sufficient conditions for the coGalois group G ( A B ) , associated to a torsion-free cover, to be trivial for an object A B in q 2 . Our results generalize those of E. Enochs and J. Rado for abelian groups.

Actions of parabolic subgroups in GL_n on unipotent normal subgroups and quasi-hereditary algebras

Thomas Brüstle, Lutz Hille (2000)

Colloquium Mathematicae

Let R be a parabolic subgroup in G L n . It acts on its unipotent radical R u and on any unipotent normal subgroup U via conjugation. Let Λ be the path algebra k t of a directed Dynkin quiver of type with t vertices and B a subbimodule of the radical of Λ viewed as a Λ-bimodule. Each parabolic subgroup R is the group of automorphisms of an algebra Λ(d), which is Morita equivalent to Λ. The action of R on U can be described using matrices over the bimodule B. The advantage of this description is that each...

Additive functions for quivers with relations

Helmut Lenzing, Idun Reiten (1999)

Colloquium Mathematicae

Additive functions for quivers with relations extend the classical concept of additive functions for graphs. It is shown that the concept, recently introduced by T. Hübner in a special context, can be defined for different homological levels. The existence of such functions for level 2 resp. ∞ relates to a nonzero radical of the Tits resp. Euler form. We derive the existence of nonnegative additive functions from a family of stable tubes which stay tubes in the derived category, we investigate when...

Additive functions on trees

Piroska Lakatos (2001)

Colloquium Mathematicae

The motivation for considering positive additive functions on trees was a characterization of extended Dynkin graphs (see I. Reiten [R]) and applications of additive functions in representation theory (see H. Lenzing and I. Reiten [LR] and T. Hübner [H]). We consider graphs equipped with integer-valued functions, i.e. valued graphs (see also [DR]). Methods are given for constructing additive functions on valued trees (in particular on Euclidean graphs) and for characterizing...

Algebras stably equivalent to trivial extensions of hereditary algebras of type à n

Zygmunt Pogorzały (1993)

Colloquium Mathematicae

The study of stable equivalences of finite-dimensional algebras over an algebraically closed field seems to be far from satisfactory results. The importance of problems concerning stable equivalences grew up when derived categories appeared in representation theory of finite-dimensional algebras [8]. The Tachikawa-Wakamatsu result [17] also reveals the importance of these problems in the study of tilting equivalent algebras (compare with [1]). In fact, the result says that if A and B are tilting...

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