Special Session 44: The theory of cluster algebras and its applications

On the acyclic quantum cluster algebras with principle coefficients

Xueqing Chen University of Wisconsin-Whitewater USA Co-Author(s):    Ming Ding, Junyuan Huang and Fan Xu

Abstract: We study a new lower bound quantum cluster algebra which is generated by the initial quantum cluster variables and the quantum projective cluster variables of an acyclic quantum cluster algebra with principle coefficients. We show that the lower bound quantum cluster algebra coincides with the corresponding acyclic quantum cluster algebra. Moreover, the dual PBW basis of this algebra is obtained. This is a joint work with M. Ding, J. Huang and F. Xu.

Fixed points of the Fomin-Zelevinsky twist

Antoine de Saint Germain the University of Hong Kong Hong Kong Co-Author(s):    Jiang-Hua Lu

Abstract: In this talk, we study fixed points of the Fomin-Zelevinsky twist of cluster algebras of finite type. We show that the Fomin-Zelevinsky twist admits a unique totally fixed point. Using this, we obtain a formula relating the exponents (which we will define) of the Fomin-Zelevinsky twist to the number of cluster variables in the underlying cluster algebra. Finally, we provide an interpretation of this result as a tropical analog of Kostant`s classical theorem relating exponents of Coxeter elements and the number of positive roots. This is based on work in progress joint with Jiang-Hua Lu

Cluster-concealed algebras and intersection matrix Lie algebras

Shengfei Geng Sichuan University Peoples Rep of China Co-Author(s):    Changjian Fu, Pin Liu

Abstract: Let $B$ be a concealed algebra, denoted by $\mathfrak{im}(B)$ the corresponding (Slowdowy) intersection matrix Lie algebra defined by the matrix of the quadratic form of $B$. Let $C=B\ltimes \text{Ext}^2_B(DB,B)$ be the corresponding cluster-concealed algebra. Based on Ringle`s work on cluster concealed algebras, it is proved that the each real Schur root of $\mathfrak{im}(B)$ can be realized by certain decomposition of some indecomposable $\tau$-rigid $C$-modules when viewed as $B$-module. Moreover, we give a bijection between the positive roots of $\mathfrak{im}(B)$ and the indecomposable $B$-modules. This is a joint work with Changjian Fu and Pin Liu.

Casimir Actions of Parabolic Positive Representations

Ivan Chi Ho Ip Hong Kong University of Science and Technology Hong Kong Co-Author(s):    Ryuichi Man, Gus Schrader

Abstract: The parabolic positive representations of $\mathcal{U}_q(\mathfrak{g}_\mathbb{R})$ were previously constructed by quantizing the classical parabolic induction corresponding to arbitrary parabolic subgroups, such that the Chevalley generators act by positive self-adjoint operators on a Hilbert space. This generalizes the (standard) positive representations introduced earlier corresponding to the minimal parabolic (i.e. Borel) subgroup. In this talk, we will show how one can study the scalar actions of the generalized Casimir operators by certain reductions from the standard representations to the parabolic cases.

On the categorifications of Goncharov--Shen`s basic triangle

Bernhard Keller Universite Paris Cite France Co-Author(s):    Miantao Liu

Abstract: With each pair consisting of a marked surface $S$ and a split simple Lie group $G$, Goncharov--Shen have associated a cluster algebra governing the higher Teichmuller space corresponding to $S$ and $G$. In the case where $S$ is a triangle, work by Miantao Liu allows to categorify this cluster algebra using the Higgs category associated to a canonical quiver with potential (constructed uniformly using a relative Calabi--Yau completion). Merlin Christ has conjectured two alternative descriptions of this Higgs category: 1) as a cosingularity category and 2) as the category of triangles in a $1$-cluster category. We will sketch a proof of his conjectures. This is a report on part of Miantao Liu`s ongoing Ph. D. thesis.

Maximal green sequences and q-characters of Kirillov-reshetikhim modules

Gleb Koshevoy IITP Russian Academy of Sciencies Russia Co-Author(s):    Yuki Kanakubo, Toshiki Nakashima

Abstract: We show that a $q$-character of a Kirillov-Reshetikhin module (KR-modules) might be obtained from a specific cluster variable of a seed obtained by applying a maximal green sequence to the initial (infinite) quiver the Hernandez-Leclerc cluster algebra. For a collection of KR-modules with nested supports, we show an explicit construction of a cluster seed which has cluster variables corresponding to the $q$-characters of KR-modules of such a collection. We prove that the product of KR-modules of such a collection is a simple module. We also construct cluster seeds with cluster variables corresponding to $q$-characters of KR-modules of some non-nested collections. We make a conjecture that tensor products of KR-modules for such non-nested collections are simple. We also show that the cluster Donaldson-Thomas transformations for double Bruhat cells can be computed the Frenkel-Mukhin algorithm and our algorithm.

Cluster symmetry and Diophantine equations

Fang Li Zhejiang University Peoples Rep of China Co-Author(s):    Fang Li

Abstract: In this talk, we will introduce the method of cluster symmetry motivated by cluster algebras and apply it to discuss the soluations of Diophantine equations.

Derived equivalences between one-branch extensions of rectangles

Yanan Lin Xiamen University Peoples Rep of China Co-Author(s):    Qiang Dong, Yanan Lin and Shiquan Ruan

Abstract: This is joint work with Qiang Dong and Shiquan Ruan. We investigate the incidence algebras arising from one-branch extensions of rectangles. There are four different ways to form such extensions, and all four kinds of incidence algebras turn out to be derived equivalent. We provide realizations for all of them by tilting complexes in a Nakayama algebra. As an application, we obtain the explicit formulas of the Coxeter polynomials for a half of Nakayama algebras.

Denominator Conjecture and string algebras

Pin Liu Southwest Jiaotong University Peoples Rep of China Co-Author(s):    Difan Deng, Changjian Fu and Shengfei Geng

Abstract: This is based on joint work with Difan Deng, Changjian Fu and Shengfei Geng. We show that most string algebras have the $\tau$-reachable property and Fomin-Zelevinsky`s denominator conjecture holds for cluster algebras of type $\mathbb{A}\mathbb{B}\mathbb{C}$.

Dual canonical bases of quantum groups

Ming Lu Sichuan University Peoples Rep of China Co-Author(s):    Xiaolong Pan

Abstract: There are two important ways to realize quantum groups, one is Hall algebra given by Ringel and Bridgeland, the other one is the convolution algebra of perverse sheaves given by Lusztig, Nakajima and Qin. In this talk, we shall compare these two realizations. The perverse sheaves give the dual canonical basis of quantum groups by Hernandez-Leclerc and Qin. We prove that dual canonical basis is invariant under braid group actions, and the transition matrix from the dual canonical basis to the basis of Hall algebra is integral and positive, which extend Lusztig`s result. We also compute the rank 1 dual canonical bases, which coincide with the double canonical bases defined by Bernstein and Greenstein, so we expect that there two bases coincide. This is joint work with Xiaolong Pan.

Additive categorification of positroid cluster structures

Matthew Pressland Universit\\\\'{e} de Caen-Normandie France Co-Author(s):

Abstract: An open positroid variety is a geometric object appearing in Postnikov`s stratification of the totally non-negative Grassmannian, and a result of Galashin and Lam is that the homogeneous coordinate ring of such a variety has two natural cluster algebra structures. Muller and Speyer conjectured a precise relationship (quasi-coincidence) between these two cluster structures, implying in particular that they define the same positive part of the variety. In this talk, I will explain how to understand the cluster structures and prove Muller and Speyer`s conjecture using the techniques of additive categorification.

Deformed 3-Calabi-Yau categories and Euclidean Artin braid groups

Yu Qiu Tsinghua University Peoples Rep of China Co-Author(s):

Abstract: We introduce a new family of quivers with potential for triangulated marked surfaces with punctures. We show that the deformation of the associated 3-Calabi-Yau categories corresponds to the partial compactification (with orbifolding) of the associated moduli spaces. As an application, we calculate the fundamental groups of these moduli spaces (of framed quadratic differentials), which in particular produces Euclidean Artin braid groups of type A, B, C and D.

Cluster algebras for Symplectic groupoid and Teichhmuller space of closed genus 2 surfaces

Michael Shapiro Michigan State University USA Co-Author(s):    L.Chekhov

Abstract: A symplectic groupoid of real unipotent upper-triangular matrices was introduced by A.Bondal. It consists of pairs of nondegenerate matrix $B$ and unipotent upper-triangular matrix $A$ such that $BAB^t$ is also unipotent upper-triangular. The symplectic groupoid possesses a natural symplectic structure that induces a natural Poisson structure on the space of unipotent upper-triangular matrices. This Poisson structure appeared earlier in papers by B.Dubrovin, M.Ugaglia, M.Mazzocco and other in relation to isomonodromic deformations. We discuss a cluster structure compatible with this Poisson structure and use it to describe a cluster structure on Teichmuller space of closed genus 2 curve.

Cluster realizations of i-quantum groups

Jinfeng Song National University of Singapore Singapore Co-Author(s):

Abstract: The i-quantum groups arising from quantum symmetric pairs are significant generalizations of quantum groups. In this talk, I will present a cluster realization of i-quantum groups, specifically an algebra embedding of an i-quantum group into a quantum cluster algebra. This embedding allows fundamental constructions of i-quantum groups to be interpreted within a cluster-theoretic framework. As an application, we derive a (dual) integral form of the i-quantum group that is invariant under braid group symmetries.

Group actions on relative cluster categories and Higgs categories

Yilin Wu University of Science and Technology of China Peoples Rep of China Co-Author(s):

Abstract: Let G be a finite group acting on an ice quiver with potential (Q, F, W). In this talk, we will discuss the associated G-action on the relative cluster category and on the Higgs category, and provide the construction of G-equivariant relative cluster category and G-equivariant Higgs category, generalizing the work of Demonet, Paquette-Schiffler, and Le Meur. In the non-simply laced case, the G-equivariant Higgs category can provide an additive categorification for cluster algebras with principal coefficients.

The multiplication formulas of quantum cluster algebras

jie xiao beijing normal university Peoples Rep of China Co-Author(s):    Zhimin Chen, Fan Xu and Fang Yang

Abstract: By applying the property of Ext-symmetry and the vector bundle structures of certain fibres, we introduce the notion of weight function and prove the multiplication formulas for weighted quantum cluster characters ( functions ) associated to abelian categories with the property of Ext-symmetry and 2-Calabi-Yau triangulated categories with cluster-tilting objects. This is joint work with Zhimin Chen, Fan Xu and Fang Yang.

Bott-Samelson atlas and Lusztig`s total positivity on a flag variety

Shizhuo Yu Nankai University Peoples Rep of China Co-Author(s):

Abstract: The Bott-Samelson atlas is an atlas on a flag variety constructed via Kazhdan-Lusztig maps. When equipping with the standard Poisson structure, the Bott-Samelson atlas makes a flag variety covered by of symmetric CGL extensions. Moreover, all shifted big cells can be realized as cut of a symmetric CGL extensions, which induce the Lusztig`s total positivity simultaneously but different cluster structures separately. In particular, each coordinate function inside the Bott-Samelson atlas is positive. This is a joint work with Jiang-Hua Lu.