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Alicja Jaworska-Pastuszak

Publications and source records attributed to Alicja Jaworska-Pastuszak.

6 recordsLinked to original sources
arXiv2026

Tame symmetric algebras of period four with small Gabriel quivers

The tame symmetric algebras of period four, TSP4 algebras for short, form an important class of algebras, with interesting links to various branches of modern algebra. The study of this class has been recently developed in two major directions. The first embraces new classes of examples of TSP4 algebras, such as virtual mutations and generalized weighted surface algebras, both extending known class of the weighted surface algebras. The second provides new classifications of TSP4 algebras (based on known results for $2$-regular case), which handle algebras, whose Gabriel quivers satisfy more general properties. An ongoing project shades a new light on the combinatorics of such algebras, introducing a new useful tool for their classification, called periodicity shadows. In this paper, we attack the problem of classification of TSP4 algebras, from another perspective, namely, we give a classification of all TSP4 algebras with not too big Gabriel quivers, i.e. having at most $5$ vertices -- but with no restrictions on their structure, as it was the case for previous classifications. The result is based on the application of the notion of periodicity shadow, which allows to compute all possible Gabriel quivers of such algebras (for small number of vertices), and recent results on interated mutations of algebras with periodic simple modules. The main result show that TSP4 algebras with Gabriel quivers having at most $5$ vertices are generalized weighted surface algebras, confirming a general conjecture in this case.

math.RT↗
arXiv2025

On Krull-Gabriel dimension of weighted surface algebras

We determine the Krull-Gabriel dimension of weighted surface algebras, a class of algebras which recently appeared in the context of classification of tame symmetric periodic algebras of non-polynomial growth. Moreover, we consider Krull-Gabriel dimension of idempotent algebras of weighted surface algebras and generalize the result in some cases.

math.RT↗
arXiv2024

On one-parameter families of hermiticity-preserving superoperators which are not positive

A one-parameter family of hermiticity-preserving superoperators is a time-dependent family $\{Φ_{t}\colon\mathbb{M}_{n}(\mathbb{C})\rightarrow\mathbb{M}_{n}(\mathbb{C})\}_{t\in\mathbb{R}}$ of hermiticity-preserving superoperators determined, in a certain sense, by real and complex polynomial functions in the variable $t\in\mathbb{R}$. The paper studies sufficient computable criteria for nonpositivity of superoperators in one-parameter families. More precisely, we give sufficient conditions for the following assertions to hold: $(1)$ every $Φ_{t}$ is not positive, $(2)$ $Φ_{t}$ is not positive for $t$ in some open interval $(u,v)\subseteq\mathbb{R}$ and $(3)$ there is some $Φ_{t}$ which is not positive. We show that in some situations $(3)$ implies $(2)$. Our approach to the problem is based on the Descartes rule of signs and the Sturm-Tarski theorem. In order to apply these facts, we introduce the sign variation formulas. These formulas are first order logical formulas in one free variable $t$, generalising sign sequences of polynomials used in Descartes rule of signs.

math-ph↗
arXiv2024

On Krull-Gabriel dimension of cluster repetitive categories and cluster-tilted algebras

Assume that $K$ is an algebraically closed field and denote by $KG(R)$ the Krull-Gabriel dimension of $R$, where $R$ is a locally bounded $K$-category (or a bound quiver $K$-algebra). Assume that $C$ is a tilted $K$-algebra and $\widehat{C},\check{C},\widetilde{C}$ are the associated repetitive category, cluster repetitive category and cluster-tilted algebra, respectively. Our first result states that $KG(\widetilde{C})=KG(\check{C})\leq KG(\widehat{C})$. Since the Krull-Gabriel dimensions of tame locally support-finite repetitive categories are known, we further conclude that $KG(\widetilde{C})=KG(\check{C})=KG(\widehat{C})\in\{0,2,\infty\}$. Finally, in the Appendix Grzegorz Bobiński presents a different way of determining the Krull-Gabriel dimension of the cluster-tilted algebras, by applying results of Geigle.

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