Abstracts
Speaker : 이신명 (Lee, Sin-Myung)
Title : Quiver representations categorifying cluster algebras: from reflection functors to quivers with potential
Abstract : While quivers and their representations have been central and fundamental objects in several branches of representation theory, they are becoming ubiquitous even beyond the representation theory itself. This talk aims to look through their developments during the first decade of the 21st century towards additive categorifications of cluster algebras, yet another omnipresent object. We will mostly follow a wonderful survey by B. Keller, "Cluster algebras, quiver representations and triangulated categories", assuming basic knowledge on quiver representations and cluster algebras (e.g. sections 2, 3 and 5.1 of the survey).
Speaker : 성정현 (Sung, Jeong Hyun)
Title : Why do we count with $q$? : From Rook Placements to Hessenberg Varieties.
Abstract : In this talk, we introduce the palindromic linked $q$-hit numbers, which arise from counting rook placements on specific board shapes. While these numbers are defined combinatorially, we address the fundamental question: "Why do we count with $q$?" We answer this by exploring the connection between these combinatorial objects and regular semisimple Hessenberg Varieties. We explain the geometric role of $q$ as the grading of the cohomology ring of the variety. And then, we discuss how the cohomology character can be expressed as a co-palindromic linear combination of these palindromic linked $q$-hit numbers. This talk is based on joint work with Seung Jin Lee.
Speaker : 천지원 (Chun, Jiwon)
Title : Cluster Categories and Cluster-Tilting Theory
Abstract : Cluster categories were introduced by Marsh, Reineke, and Zelevinsky as a categorical framework for the categorification of the essential ingredients of cluster algebras, such as clusters and cluster variables, using quiver representations. They are constructed as certain orbit categories of the derived categories of quiver representations of finite-dimensional hereditary algebras. Cluster-tilting theory extends the classical tilting theory of hereditary algebras to the setting of cluster categories. In this framework, mutations of tilting objects naturally categorify the mutation process in cluster algebras. On the other hand, Geiss, Leclerc, and Schröer showed that (quantum) unipotent coordinate rings admit a cluster algebra structure via cluster-tilting theory for preprojective algebras. In this talk, we survey these developments and present concrete examples to get intuition for cluster categories and cluster-tilting theory.
Speaker : 정우석 (Jung, Woo-Seok)
Title : Comparing cluster structures on subalgebras of bosonic extensions and half decorated double Bott-Samelson varieties
Abstract : For a positive braid word $\mathtt{b}$, we compare cluster structures on the subalgebra $\widehat{\mathcal{A}}(\mathtt{b})$ of a bosonic extension and the coordinate ring of the half decorated double Bott-Samelson variety $\mathrm{Conf}(\mathtt{b})$. The algebra $\widehat{\mathcal{A}}(\mathtt{b})$ can be viewed as a quantum deformation of the Grothendieck ring of a distinguished subcategory of the module category of a quantum affine algebra, and its quantum cluster structure was constructed by Kashiwara-Kim-Oh-Park using admissible chains of i-boxes. On the other hand, cluster structures on braid varieties (which include $\mathrm{Conf}(\mathtt{b})$) were investigated by Casals-Gorsky-Gorsky-Le-Shen-Simental via Demazure weaves. While the specialization of $\widehat{\mathcal{A}}(\mathtt{b})$ at $q=1$ is isomorphic to the coordinate ring of $\mathrm{Conf}(\mathtt{b})$ up to localization, their seeds are described by these distinct combinatorial data. We establish a direct dictionary interpreting admissible chains of i-boxes as Demazure weaves, thereby unifying the two cluster structures. We will also discuss applications of this correspondence. This is ongoing joint work with Jisun Huh, Myungho Kim, and Euiyong Park.
Speaker : 오동건 (Oh, Donggeon)
Title : An Introduction to Ribbon Categories
Abstract : Ribbon categories bridge the gap between representation theory and knot theory. This talk provides a comprehensive overview of ribbon categories, ranging from their rigorous definitions to their diagrammatic intuition. We begin by establishing the categorical hierarchy, progressing from monoidal categories through richer structures to formally define ribbon categories. To clarify these abstract concepts, we consider the category of finite-dimensional modules over a ribbon Hopf algebra as a concrete example. Finally, we discuss the graphical calculus of ribbon categories and construct link invariants.
Speaker : 윤상원 (Yoon, Sangwon)
Title : Lie algebra and superconformality of vertex algebras
Abstract : In the case of Lie superalgebras, supersymmetry can exist at the Lie algebraic level. However, affine vertex algebras, an important class of vertex algebras, do not in general possess superconformality, even when they are associated with Lie superalgebras. Here, superconformality refers to a specific notion of supersymmetry in the context of vertex algebras. Accordingly, a well-known class that always incorporates superconformality is the class of super-affine vertex algebras. These algebras carry N=1 superconformality, for any Lie (super)algebra, and we will review this construction. We will then explain how a special structure on Lie algebras gives rise to additional superconformal structures in super-affine vertex algebras.
Speaker : 이재현 (Lee, Jaehyun)
Title : An Introduction to Modular Tensor Category with Example from Lattice VOA Module
Abstract : Modular tensor category(MTC) is a semisimple ribbon category with some additional properties. Multiplicity coefficients can be easily computed by Verlinde formula in modular tensor category. In order to give an example of modular tensor category, I introduce a slightly different definition of vertex operator algebra(VOA) module and their tensor product, called as the fusion product. Category of certain VOA module with fusion product becomes a modular tensor category. In this talk, I give a specific example of this: lattice VOA module of \mathfrak{sl_2}
Speaker : 이승빈 (Lee, Seungbeen)
Title : Link invariants from quantum groups of Type A
Abstract :It is a celebrated result by Reshetikhin and Turaev that any ribbon category yields invariants of links. A fundamental example of such a category is the category of finite-dimensional modules over the quantum group $U_q(\mathfrak{g})$ for a simple Lie algebra $\mathfrak{g}$. In this talk, focusing on the case of Type A, we discuss link invariants derived from the quantum group $U_q(\mathfrak{sl}_n)$ and how these invariants recover classical link invariants, such as the Jones polynomial.