Related papers: A categorification of acyclic principal coefficien…

In 2009, Keller and Yang categorified quiver mutation by interpreting it in terms of equivalences between derived categories. Their approach was based on Ginzburg's Calabi-Yau algebras and on Derksen-Weyman-Zelevinsky's mutation of quivers…

We describe a reduction technique for stably 2-Calabi--Yau Frobenius extriangulated categories $\mathcal{F}$ with respect to a functorially finite rigid subcategory $\mathcal{X}$. The reduction of such a category is another category…

We show that the class of twisted fractionally Calabi-Yau algebras of finite global dimension coincides with the stable endomorphism algebras of $d$-cluster tilting modules over $d$-representation-finite algebras. This is an application of…

We show that locally acyclic cluster algebras have (at worst) canonical singularities. In fact, we prove that locally acyclic cluster algebras of positive characteristic are strongly F-regular. In addition, we show that upper cluster…

Building on classical invariant theory, it is observed that the polarised traces generate the centraliser $Z_L(sl(N))$ of the diagonal embedding of $U(sl(N))$ in $U(sl(N))^{\otimes L}$. The paper then focuses on $sl(3)$ and the case $L=2$.…

We establish the foundations of categorical weave calculus, developing the diagrammatic calculus of weaves and braid varieties within the study of Calabi-Yau triangulated categories and cluster tilting theory. This is achieved by…

We consider an orbit category of the bounded derived category of a path algebra of type A_n which can be viewed as a -(m+1)-cluster category, for m >= 1. In particular, we give a characterisation of those maximal m-rigid objects whose…

We study rooted cluster algebras and rooted cluster morphisms which were introduced in \cite{ADS13} recently and cluster structures in $2$-Calabi-Yau triangulated categories. An example of rooted cluster morphism which is not ideal is…

We present a unified apoach to the study of separable and Frobenius algebras. The crucial observation is thsat both cases are related to the nonlinear equation $R^{12}R^{23}=R^{23}R^{13}=R^{13}R^{12}$, called the FS-equation. Given a…

Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the associated preprojective algebra. To each terminal representation M of Q (these are certain preinjective representations), we attach a natural subcategory $C_M$ of…

We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers…

This is an introductory survey on cluster algebras and their (additive) categorification using derived categories of Ginzburg algebras. After a gentle introduction to cluster combinatorics, we review important examples of coordinate rings…

We survey results on mutations of Jacobian algebras, while simultaneously extending them to the more general setup of frozen Jacobian algebras, which arise naturally from dimer models with boundary and in the context of the additive…

We propose a new framework for categorifying skew-symmetrizable cluster algebras. Starting from an exact stably 2-Calabi-Yau category C endowed with the action of a finite group G, we construct a G-equivariant mutation on the set of maximal…

We show that the bigroupoid of separable symmetric Frobenius algebras over an algebraically closed field and the bigroupoid of finitely semi-simple Calabi-Yau categories are equivalent. To this end, we construct a trace on the category of…

In previous work, the first three authors conjectured that the ring of regular functions on a natural class of affine log Calabi-Yau varieties (those with maximal boundary) has a canonical vector space basis parameterized by the integral…

We define the class of rigid Frobenius algebras in a (non-semisimple) modular category and prove that their categories of local modules are, again, modular. This generalizes previous work of A. Kirillov, Jr. and V. Ostrik [Adv. Math. 171…

The period geometry of Calabi-Yau $n$-folds, characterised by their variations of Hodge structure governed by Griffiths transversality, a graded Frobenius algebra, an integral monodromy and an intriguing arithmetic structure, is analysed…

We develop a general theory of cluster categories, applying to a 2-Calabi-Yau extriangulated category $\mathcal{C}$ and cluster-tilting subcategory $\mathcal{T}$ satisfying only mild finiteness conditions. We show that the structure theory…

The article gives a ring theoretic perspective on cluster algebras. Gei{\ss}-Leclerc-Schr\"oer prove that all cluster variables in a cluster algebra are irreducible elements. Furthermore, they provide two necessary conditions for a cluster…