Related papers: Labelled seeds and the mutation group

We construct an injective algebra homomorphism of the quantum group $U_q(\mathfrak{sl}_{n+1})$ into a quantum cluster algebra $\mathbf{L}_n$ associated to the moduli space of framed $PGL_{n+1}$-local systems on a marked punctured disk. We…

We introduce signed exceptional sequences as factorizations of morphisms in the cluster morphism category. The objects of this category are wide subcategories of the module category of a hereditary algebra. A morphism $[T]:\mathcal A\to…

Modular functors are traditionally defined as systems of projective representations of mapping class groups of surfaces that are compatible with gluing. They can formally be described as modular algebras over central extensions of the…

We show that the quantum coordinate ring of the unipotent subgroup N(w) of a symmetric Kac-Moody group G associated with a Weyl group element w has the structure of a quantum cluster algebra. This quantum cluster structure arises naturally…

In 2003, Fomin and Zelevinsky obtained Cartan-Killing type classification of all cluster algebras of finite type, i.e. cluster algebras having only finitely many distinct cluster variables. A wider class of cluster algebras is formed by…

Let Q be a finite quiver without oriented cycles, and let $\Lambda$ be the corresponding preprojective algebra. Let g be the Kac-Moody Lie algebra with Cartan datum given by Q, and let W be its Weyl group. With w in W is associated a…

For a rooted cluster algebra $\mathcal{A}(Q)$ over a valued quiver $Q$, a \emph{symmetric cluster variable} is any cluster variable belonging to a cluster associated with a quiver $\sigma (Q)$, for some permutation $\sigma$. The subalgebra…

We introduce a pseudo $\mathbb{N}$-grading on the cluster auotmorphism group $\operatorname{Aut}(\mathcal{A})$ with respect to an initial seed of $\mathcal{A}$, which consists of a family of subsets $\{G_i\}_{i\in \mathbb{N}}$ of…

We extend based cluster algebras from the finite rank case to the infinite rank case. By extending (quantum) cluster algebras whose initial seeds are associated with signed words (arising from double Bott--Samelson cells), we recover…

Let $G$ be a finite group acting on an ice quiver with potential $(Q, F, W)$. We construct the corresponding $G$-equivariant relative cluster category and $G$-equivariant Higgs category, extending the work of Demonet. Using the orbit…

We characterize mutation-finite cluster algebras of rank at least 3 using positive semi-definite quadratic forms. In particular, we associate with every unpunctured bordered surface a positive semi-definite quadratic space $V$, and with…

We introduce a framework for $\mathbb{Z}$-gradings on cluster algebras (and their quantum analogues) that are compatible with mutation. To do this, one chooses the degrees of the (quantum) cluster variables in an initial seed subject to a…

With every reduced $E$-Fountain semigroup $S$ which satisfies the generalized right ample condition we associate a category with zero morphisms $\mathcal{C}(S)$. Under some assumptions we prove an isomorphism of $\Bbbk$-algebras $\Bbbk…

We introduce and study "2-roots", which are symmetrized tensor products of orthogonal roots of Kac--Moody algebras. We concentrate on the case where $W$ is the Weyl group of a simply laced Y-shaped Dynkin diagram $Y_{a,b,c}$ having $n$…

We consider the symplectic groupoid of pairs $(B,\mathbb{A})$ with $\mathbb A$ unipotent upper-triangular matrices and $B\in GL_n$ being such that $\widetilde {\mathbb A}=B{\mathbb A} B^{\text{T}}$ are also unipotent upper-triangular…

The paper is motivated by an analogy between cluster algebras and Kac-Moody algebras: both theories share the same classification of finite type objects by familiar Cartan-Killing types. However the underlying combinatorics beyond the two…

Let $(W, S)$ be a Coxeter system equipped with a fixed automorphism $\ast$ of order $\leq 2$ which preserves $S$. Lusztig (and with Vogan in some special cases) have shown that the space spanned by set of "twisted" involutions was naturally…

We study the cluster automorphism group $Aut(\mathcal{A})$ of a coefficient free cluster algebra $\mathcal{A}$ of finite type. A cluster automorphism of $\mathcal{A}$ is a permutation of the cluster variable set $\mathscr{X}$ that is…

We study consequences of a monoidal categorification of the unipotent quantum coordinate ring $A_q(\mathfrak{n}(w))$ together with the Laurent phenomenon of cluster algebras. We show that if a simple module $S$ in the category $\mathcal…

This article tries to generalize former works of Derksen, Weyman and Zelevinsky about skew-symmetric cluster algebras to the skew-symmetrizable case. We introduce the notion of group species with potentials and their decorated…