Related papers: Introduction to Cluster Algebras. Chapters 4-5
We initiate the study of cluster algebras in Feynman integrals in dimensional regularization. We provide evidence that four-point Feynman integrals with one off-shell leg are described by a $C_{2}$ cluster algebra, and we find cluster…
We describe a new way to relate an acyclic, skew-symmetrizable cluster algebra to the representation theory of a finite dimensional hereditary algebra. This approach is designed to explain the c-vectors of the cluster algebra. We obtain a…
We review several constructions of integrable systems with an underlying cluster algebra structure, in particular the Gekhtman-Shapiro-Tabachnikov-Vainshtein construction based on perfect networks and the Goncharov-Kenyon approach based on…
Given a finite dimensional algebra $C$ (over an algebraically closed field) of global dimension at most two, we define its relation-extension algebra to be the trivial extension $C\ltimes \Ext_C^2(DC,C)$ of $C$ by the $C$-$C$-bimodule…
We give the complete algebraic classification of all complex 4-dimensional nilpotent algebras. The final list has 234 (parametric families of) isomorphism classes of algebras, 66 of which are new in the literature.
This article is a survey of algebra in the $\infty$-categorical context, as developed by Lurie in "Higher Algebra", and is a chapter in the "Handbook of Homotopy Theory". We begin by introducing symmetric monoidal stable…
Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field $k$ and $\mathscr{C}_{F^m}$ be the repetitive cluster category of $H$ with $m\geq 1$. We investigate the properties of cluster tilting objects in…
We classify self-injective radical cube zero algebras with respect to whether they satisfy certain finite generation conditions sufficient to have a fruitful theory of support varieties defined via Hochschild cohomology in the vein of…
We complete the description of group gradings on finite-dimensional incidence algebras. Moreover, we classify the finite-dimensional graded algebras that can be realized as incidence algebras endowed with a group grading.
The explosion in the amount of data available for analysis often necessitates a transition from batch to incremental clustering methods, which process one element at a time and typically store only a small subset of the data. In this paper,…
We compute the class group of a full rank upper cluster algebra in terms of its exchange polynomials. As a corollary, we recover a theorem by Cao, Keller, and Qin from 2023 characterizing the UFDs among these algebras. Furthermore, under…
Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. To a cluster algebra of simply laced Dynkin type one can associate the cluster category. Any cluster of the cluster algebra corresponds…
Classification theory of elementary classes deals with first order (elementary) classes of structures (i.e. fixing a set T of first order sentences, we investigate the class of models of T with the elementary submodel notion). It tries to…
In this paper we give a graph theoretic combinatorial interpretation for the cluster variables that arise in most cluster algebras of finite type. In particular, we provide a family of graphs such that a weighted enumeration of their…
Holm and Jorgensen have shown the existence of a cluster structure on a certain category $D$ that shares many properties with finite type $A$ cluster categories and that can be fruitfully considered as an infinite analogue of these. In this…
Let $F$ be a finite field of characteristic $p$. The structures of the unit groups of group algebras over $F$ of the three groups $D_{24}$, $S_4$ and $SL(2, \mathbb{Z}_3)$ of order $24$ are completely described in \cite{K4, SM, SM1, FM,…
We compute the Hochschild cohomology groups of the cluster-tilted algebras of finite representation type.
We study the module categories of a tilted algebra C and the corresponding cluster-tilted algebra B. In particular, we study which $\tau$-rigid C-modules are also $\tau$-rigid B-modules.
We develop new techniques to classify basic algebras of blocks of finite groups over algebraically closed fields of prime characteristic. We apply these techniques to simplify and extend previous classifications by Linckelmann, Murphy and…
Clustering algorithms aim to organize data into groups or clusters based on the inherent patterns and similarities within the data. They play an important role in today's life, such as in marketing and e-commerce, healthcare, data…