### A categorification of the square root of -1

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.

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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.

In continuation of our article in Colloq. Math. 116.1, we give a complete description of the symmetric algebras of strictly canonical type by quivers and relations, using Brauer quivers.

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 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 $A\in \u2099\left(\mathbb{Z}\right)$ is ℤ-congruent to its transpose ${A}^{tr}$ is also discussed. An affirmative answer is given for the incidence matrices ${C}_{I}$ and the Tits matrices $C{\u0302}_{I}$ of positive one-peak posets I.

We investigate the category $\text{mod}\Lambda $ of finite length modules over the ring $\Lambda =A{\otimes}_{k}\Sigma $, where $\Sigma $ 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 ${P}_{j}=A\otimes {E}_{j}$. By a result of K. Fuller, ${P}_{j}$ induces a category equivalence from which we deduce that $\text{mod}\Lambda \simeq {\coprod}_{j}badhbox{P}_{j}$. As a consequence we can (1) construct for each elementary $k$-algebra $A$ over a finite field $k$ a nonartinian noetherian ring $\Lambda $ such that $\text{mod}A\simeq \text{mod}\Lambda $, (2) find twisted...

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 ${q}_{\omega}$ consisting of the representations of an infinite line quiver. This generalization of Baer’s Lemma...

In this paper we define an action of the Weyl group on the quiver varieties ${M}_{m,\lambda}\left(v\right)$ with generic $(m,\lambda )$.

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 $R=\mathbb{Z},$ we find necessary and sufficient conditions for the coGalois group $G(A\u27f6B),$ associated to a torsion-free cover, to be trivial for an object $A\u27f6B$ in ${q}_{2}.$ Our results generalize those of E. Enochs and J. Rado for abelian groups.

In this article we characterize those abelian groups for which the coGalois group (associated to a torsion free cover) is equal to the identity.

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 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...

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...

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...