Special Session 109: Cluster Algebras, Hall Algebras and Their Applications

Discreteness for the derived category of a graded skew-gentle algebra

wen Chang Shaanxi Normal University Peoples Rep of China Co-Author(s):    HAIBO JIN, SIBYLLE SCHROLL, QI WANG

Abstract: I will characterize silting-discreteness for the perfect derived categories of graded gentle and skew-gentle algebras. Specifically, for a graded gentle algebra, silting-discreteness is equivalent to its associated surface being of genus zero with non-zero winding numbers for all simple closed curves. We further extend this geometric characterization to graded skew-gentle algebras via orbifold surface models.

Fundamental relations in quantum cluster algebras

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

Abstract: Let A_q be an arbitrary quantum cluster algebra with principal coefficients. We give the fundamental relations between the quantum cluster variables arising from one-step mutations from the initial cluster in A_q. Immediately and directly, we obtain an algebra homomorphism from the corresponding (untwisted) quantum group to A_q. This is a joint work with J. Huang, M. Ding and F. Xu.

Sign-coherence and tropical sign pattern for rank 3 real cluster-cyclic exchange matrices

Zhichao Chen University of Science and Technology of China Peoples Rep of China Co-Author(s):    Ryota Akagi

Abstract: We study the sign-coherence of $c$-vectors for rank $3$ real cluster-cyclic skew-symmetrizable cluster algebras. The sign-coherence property was conjectured by Fomin-Zelevinsky and proved in the integer skew-symmetrizable case by Gross-Hacking-Keel-Kontsevich. We extend this result to the rank $3$ real cluster-cyclic setting. In addition, we establish a self-contained recursion and a monotonicity property for these $c$-vectors, and show that they arise as roots of certain quadratic equations. As applications, we prove that the exchange graphs of the associated $C$-patterns and $G$-patterns are $3$-regular trees. We also investigate tropical sign patterns and realize the dihedral group $\mathcal{D}_6$ via cluster mutations. This is a joint work with Ryota Akagi.

Representations of bound quivers over the virtual field and their Ringel-Hall algebras

Changjian FU Sichuan University Peoples Rep of China Co-Author(s):

Abstract: The theory of quiver representations over the virtual field $\mathbb{F}_1$ can be naturally extended to bound quivers involving monomial and commutativity relations. We further investigate Tits`s philosophy on $\mathbb{F}_1$ by examining the Ringel-Hall algebras associated with specific classes of bound quivers, including Nakayama bound quivers and gentle one-cycle quivers $\Lambda(n-1, 1, 1)$. Our findings extend and refine the results of Szczesny for quivers of type $A_n$ and cyclic quivers.

Ring theoretic approach to double centralizer property and applications

JUN HU Beijing Institute of Technology Peoples Rep of China Co-Author(s):

Abstract: The double centralizer property is one of the most fundamental principles in Lie theory and representation theory. It underpins many classical results such as Schur-Weyl duality, which connects the representations of general linear Lie algebras and symmetric groups via mutual centralization on tensor spaces. I will introduce a ring theoretic approach (by Auslander and Solberg, K{\o}nig, Slung{\aa}rd and Xi, etc) to the proof of the double centralizer property in many examples, and I will talk about some of their generalizations and applications.

The quantum necklace Lie algebra and the HOMFLYPT skein algebra

Hiroaki Karuo Gakushuin University Japan Co-Author(s):    Shunsuke Tsuji

Abstract: In the classical setting, it is known that the necklace Lie algebra and a graded quotient of the Goldman Lie algebra are isomorphic. This is an example of intersection between 2-dimensional topology and representation theory. There are quantizations of the necklace Lie algebra and the Goldman Lie algebra, called the quantum necklace Lie algebra and the HOMFLYPT skein algebra respectively. However, we did not know any relation between them as the classical setting. In this talk, we define an appropriate filtration on the HOMFLYPT skein algebra and construct an isomorphism between the quantum necklace Lie algebra and the graded quotient of the HOMFLYPT skein algebra. This gives us interactive applications, e.g., a topological interpretation of the quantum necklace Lie algebra and bases of the graded quotient of the HOMFLYPT skein algebra. This is a joint work with Shunsuke Tsuji (Meiji University). If time permits, I will explain the formality problem of the HOMFLYPT skein algebra related to the Kashiwara--Vergne problem.

Exceptional curves and quantum loop algebras

Ming Lu Sichuan University Peoples Rep of China Co-Author(s):    Zhongwei Fang, Shiquan Ruan

Abstract: Schiffmann and Dou-Jiang-Xiao use the Hall algebras of weighted projective lines to realize quantum loop algebras of simply-laced type. In this talk, we generalize their results, and use the Hall algebras of exceptional curves to realize the quantum loop algebras associated to the corresponding valued star-shaped graphs. In particular, we are able to realize (twisted and untwisted) quantum affine algebras in Drinfeld new presentation.

Presentation of rational Schur algebras

Frantisek Marko The Pennsylvania State University USA Co-Author(s):    Frantisek Marko

Abstract: The rational Schur algebra S(n,r,s) over an arbitrary ground field K is represented as a quotient of the distribution algebra Dist(G) of the general linear group G=GL(n) by an ideal I(n,r,s). We provide an explicit description of the generators of I(n,r,s). Over fields K of characteristic zero, we complete a presentation of S(n,r,s) in terms of generators and relations originally considered by Dipper and Doty, and then followed up by Donkin. The explicit presentation over ground fields of positive characteristics is new.

Modulated graphs with potentials and skew-symmetrizable cluster algebras

Lang Mou UC Davis USA Co-Author(s):

Abstract: Cluster algebras in the skew-symmetric case admit a powerful categorification through representations of quivers with potentials, following the work of Derksen-Weyman-Zelevinsky. Extending this picture to the skew-symmetrizable case remains an important open problem. Modulated graphs with potentials generalize quivers with potentials and thus provide a promising candidate for such an extension. In this talk, I will present a mutation theory for modulated graphs with potentials and discuss how their representations may be used to categorify certain skew-symmetrizable cluster algebras. The goal is to suggest a representation-theoretic framework that parallels the skew-symmetric theory while capturing new phenomena arising in the skew-symmetrizable setting.

Skein algebras on punctured surfaces

Linhui Shen Michigan State University USA Co-Author(s):    Zhe Sun, Daping Weng

Abstract: We begin with a concrete example of Poisson structures associated with the cluster A-algebra from the Markov quiver. Although its Poisson bracket is not log canonical with respect to any cluster seed, it is nevertheless compatible with cluster mutations. We generalize this Poisson structure to the Fock-Goncharov moduli space A_{G,S} associated with punctured surfaces. When G is SL3, we construct a quantization of these Poisson structures, providing a natural generalization of the Roger-Yang Skein algebra. This is joint work in progress with Zhe Sun and Daping Weng.

Webs and their intersections

Zhe Sun University of Science and Technology of China Peoples Rep of China Co-Author(s):    Daniel Douglas, Linhui Shen, Daping Weng, Zhihao Wang, Tsukasa Ishibashi, Wataru Yuasa

Abstract: G-Webs as certain trivalent graphs on the surfaces, appear naturally in the G-skein algebra and its classical limit the regular function ring of the G character variety. I will explain how we introduce their intersection number to parameterize a basis of the algebra for SL3 (joint work with Daniel Douglas, Linhui Shen, Daping Weng, Zhihao Wang) and Sp4 (joint work with Tsukasa Ishibashi, Wataru Yuasa).

Quantum cluster realization for projected stated SLn-skein algebras

Zhihao Wang Korea Institute for Advanced Study Peoples Rep of China Co-Author(s):    Min Huang

Abstract: In this talk, I will review the definition of the (reduced) stated ${\rm SL}_n$-skein algebra associated to a punctured bordered surface (or marked surface). I will then introduce a quantum cluster algebra structure on the skew field of fractions of the reduced stated ${\rm SL}_n$-skein algebra in the case where the surface has no interior punctures. Furthermore, I will show that the reduced stated ${\rm SL}_n$-skein algebra embeds into the corresponding quantum cluster algebra when each connected component of the surface contains at least two punctures. In particular, every stated arc is a cluster variable up to multiplication by frozen variables. This is joint work with Min Huang.

Boundary Measurement Maps on Legendrian Weaves

Daping Weng University of North Carolina at Chapel Hill USA Co-Author(s):    Yoon Jae Nho

Abstract: Boundary measurement maps were first introduced by Postnikov as a tool to parametrize positroid cells in Grassmannians. They depend on the choices of a reduced plabic graph and a perfect orientation of the graph. In my dissertation, I showed that the boundary measurement maps of the top cells of Grassmannians compute the F-polynomials for DT transformations. In recent years, a new combinatorial tool called Legendrian weave was introduced by Casals and Zaslow, and it can be used to describe cluster structures on braid varieties. In this talk, I will define a new family of boundary measurement maps on Legendrian weaves, which parametrize (an open torus of) the flag moduli space in terms of weighted cycles on weaves. I will discuss how to use these boundary measurement maps to recover F-polynomials for DT transformations and their connection to the spectral networks of Gaiotto-Moore-Neitzke. This is based on joint work in progress with Yoon Jae Nho.

Root categories and Lie groups

Jie Xiao Beijing Normal University Peoples Rep of China Co-Author(s):

Abstract: D. Happel introduced the root category as a two-periodic orbit triangulated category R of the derived category of Dynkin quiver. The Gabriel Theorem can be stated with the Auslander-Reiten quiver of R, not only for the positive roots $\Phi^+$ but also the whole root system Phi. We introduce here a process to build up semi-simple Lie algebras and Chevalley groups via Hall algebra approach. The construction can be applied to a realization of compact real form and maximal compact subgroups from the root category R, and obtain the Peter-Weyl Theorem and the Plancherel Theorem for compact groups. This is a joint work with Buyan Li.

Skein and cluster algebras of unpunctured surfaces for rank $2$ simple Lie algebras

Wataru Yuasa Hiroshima University Japan Co-Author(s):    Tsukasa Ishibashi

Abstract: This talk surveys recent developments in the relation between skein algebras and quantum cluster algebras for rank-two simple Lie algebras $\mathfrak{g}=\mathfrak{sl}_3,\mathfrak{sp}_4,\mathfrak{g}_2$. I will explain how $\mathfrak{g}$-webs on unpunctured marked surfaces are related to quantum cluster algebras associated with moduli spaces of decorated local systems on the surface. The main goal of the talk is to present the general picture, in particular, how one constructs embeddings of skein algebras into the corresponding quantum cluster algebras. I will also present explicit examples of $\mathfrak{g}$-webs corresponding to cluster variables.