Related papers: Quasi-cluster algebras from non-orientable surface…
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 introduce a class of non-commutative algebras that carry a non-commutative (geometric) cluster structure which are generated by identical copies of generalized Weyl algebras. Equivalent conditions for the finiteness of the set of the…
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 develop a version of cluster algebra extending the ring of Laurent polynomials by adding Grassmann variables. These algebras can be described in terms of `extended quivers' which are oriented hypergraphs. We describe mutations of such…
We continue the exploration of various appearances of cluster algebras in scattering amplitudes and related topics in physics. The cluster configuration spaces generalize the familiar moduli space ${\mathcal M}_{0,n}$ to finite-type cluster…
Motivated by the study of (m,n)-quasitilted algebras, which are the piecewise hereditary algebras obtained from quasitilted algebras of global dimension two by a sequence of (co)tiltings involving n-1 tilting modules and m-1 cotilting…
The cluster-tilted algebras have been introduced by Buan, Marsh and Reiten, they are the endomorphism rings of cluster-tilting objects $T$ in cluster categories; we call such an algebra cluster-concealed in case $T$ is obtained from a…
We give a uniform geometric realization for the cluster algebra of an arbitrary finite type with principal coefficients at an arbitrary acyclic seed. This algebra is realized as the coordinate ring of a certain reduced double Bruhat cell in…
We introduce a new class of algebras, which we call cluster-tilted. They are by definition the endomorphism algebras of tilting objects in a cluster category. We show that their representation theory is very close to the representation…
Cluster ensemble is a pair of positive spaces (X, A) related by a map p: A -> X. It generalizes cluster algebras of Fomin and Zelevinsky, which are related to the A-space. We develope general properties of cluster ensembles, including its…
Cluster algebras are a class of commutative algebras whose generators are defined by a recursive process called mutation. We give a brief introduction to cluster algebras, and explain how discrete integrable systems can appear in the…
We classify the pairs $(C,G)$ where $C$ is a seminormal curve over an arbitrary field $k$ and $G$ is a smooth connected algebraic group acting faithfully on $C$ with a dense orbit, and we determine the equivariant Picard group of $C$. We…
In this paper, we show that, for skew-symmetric cluster algebras, the c-vectors of any seed with respect to an acyclic initial seed define a quasi-Cartan companion of the corresponding exchange matrix. As an application, we show that any…
Quasi-Boolean algebras were introduced as the generalization of Boolean algebras in the setting of quantum computation logic. In this paper, we investigate the completeness and congruences of quasi-Boolean algebras. First, we discuss the…
We express cluster variables of type $B_n$ and $C_n$ in terms of cluster variables of type $A_n$. Then we associate a cluster tilted bound symmetric quiver $Q$ of type $A_{2n-1}$ to any seed of a cluster algebra of type $B_n$ and $C_n$.…
We provide a concrete realization of the cluster algebras associated with Q-systems as amalgamations of cluster structures on double Bruhat cells in simple algebraic groups. For nonsimply-laced groups, this provides a cluster-algebraic…
We define a new family of noncommutative generalizations of cluster algebras called polygonal cluster algebras. These algebras generalize the noncommutative surfaces of Berenstein-Retakh, and are inspired by the emerging theory of…
We study cluster algebras with principal coefficient systems that are associated to unpunctured surfaces. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of perfect…
We study $g$-vector cones associated with clusters of cluster algebras defined from a marked surface $(S,M)$ of rank $n$. We determine the closure of the union of $g$-vector cones associated with all clusters. It is equal to $\mathbb{R}^n$…
Establishing whether an algebra is quasi-hereditary or not is, in general, a difficult problem. In this paper we introduce a sufficient criterion to determine whether a general finite dimensional algebra is quasi-hereditary by showing that…