Related papers: Deformation Theory for Finite Cluster Complexes
We study cluster algebras that are associated to unpunctured surfaces with coefficients arising from boundary arcs. We give a direct formula for the Laurent polynomial expansion of cluster variables in these cluster algebras in terms of…
We generalize the Caldero-Chapoton formula for cluster algebras of finite type to the skew-symmetrizable case. This is done by replacing representation categories of Dynkin quivers by categories of locally free modules over certain…
We mainly introduce an abstract pattern to study cluster algebras. Cluster algebras, generalized cluster algebras and Laurent phenomenon algebras are unified in the language of generalized Laurent phenomenon algebras (briefly, GLP algebras)…
We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and…
Let $F$ be a CM field and let $(\overline{r}_{\pi,\lambda})_{\lambda}$ be the compatible system of residual $\mathcal{G}_n$-valued representations of $\operatorname{Gal}_{F}$ attached to a RACSDC automorphic representation $\pi$ of…
In this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of type $\widetilde{A}$. We are particularly interested…
Comparing the module categories of an algebra and of the endomorphism algebra of a given support $\tau$-tilting module, we give a generalization of the Brenner-Butler's tilting theorem in the framework of $\tau$-tilting theory. Afterwards…
In this work, we study the deformation theory of $\cE_n$-rings and the $\cE_n$ analogue of the tangent complex, or topological Andr\'e-Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber…
The main purpose of this article is to develop an explicit derived deformation theory of algebraic structures at a high level of generality, encompassing in a common framework various kinds of algebras (associative, commutative, Poisson...)…
We extend recent work of the third author and Kouloukas by constructing deformations of integrable cluster maps corresponding to the Dynkin types $A_{2N}$, lifting these to higher-dimensional maps possessing the Laurent property and…
The article concerns the subalgebra U_v^+(w) of the quantized universal enveloping algebra of the complex Lie algebra sl_{n+1} associated with a particular Weyl group element of length 2n. We verify that U_v^+(w) can be endowed with the…
A perturbative expansion of knot invariants is derived using quantum cluster algebras. By interpreting the $R$-matrix of $U_q(\mathfrak{sl}_2)$ as a cluster transformation and introducing an auxiliary parameter $\epsilon$, we derive a…
We consider the isomonodromic deformations of irregular-singular connections defined on principal bundles over complex curves: for any complex reductive structure group G, and any polar divisor; allowing for a twisted/ramified formal normal…
Let $\CC$ be a Hom-finite triangulated 2-Calabi-Yau category with a cluster-tilting object $T$. Under a constructibility condition we prove the existence of a set $\mathcal G^T(\CC)$ of generic values of the cluster character associated to…
In this paper, we present a unified perspective on sphere consistent truncations based on the classical geometric properties of sphere bundles. The backbone of our approach is the global angular form for the sphere. A universal formula for…
We prove that in the skew-symmetrizable cluster algebras associated by Felikson-Shapiro-Tumarkin to unpunctured surfaces with orbifold points of order $2$ and a specific choice of weights, the Laurent expansion of any cluster variable with…
With any non necessarily orientable unpunctured marked surface (S,M) we associate a commutative algebra, called quasi-cluster algebra, equipped with a distinguished set of generators, called quasi-cluster variables, in bijection with the…
In this paper we investigate the endomorphism algebras of standard cluster tilting objects in the stably 2-Calabi-Yau categories $\Sub{\Lambda_w}$ with elements $w$ in Coxeter groups in \cite{BIRSc}. They are examples of the 2-Auslander…
This work presents a generalization of derived blow-ups and of the derived deformation to the normal bundle from derived algebraic geometry to any geometric context. The latter is our proposed globalization of a derived algebraic context,…
The weak distortions of high-redshift galaxies caused by gravitational light deflection near clusters of galaxies can be used to reconstruct the projected (two-dimensional) surface mass density of intermediate redshift clusters. This…