Titles & Abstracts

Speaker : 서의린 (Seoul National University)

Title : Problems on Lie superalgebras arising from W-algebras theory
Abstract : A W-algebra is defined for a Lie superalgebra and an even nilpotent element with certain properties. Hence the theory of W-algebras has been developed based on the theory of Lie superalgebras. Recently, I have been working on a new object called SUSY W-algebra, which is a supersymmetric analogue of a W-algebra. It can be also constructed for a Lie superalgebra for an odd nilpotent element. One can expect some new observations are needed to understand this new algebra. In this lecture, I will explain what are interesting problems on Lie superalgebras that can be applied to W-algebras theory.

Speaker : 허태혁 (Seoul National University)

Title :  Title: Extremal weight crystals over affine Lie algebras of infinite rank

Abstract : We explain extremal weight crystals over affine Lie algebras of infinite rank using combinatorial models: a spinor model due to Kwon, and an infinite rank analogue of Kashiwara-Nakashima tableaux due to Lecouvey. In particular, we show that Lecouvey's tableau model is isomorphic to an extremal weight crystal of level zero. Using these combinatorial models, we explain an algebra structure of the Grothendieck ring for a category consisting of some extremal weight crystals.

Speaker : 최동준 (Seoul National University)

Title : Introduction to Lie superalgebras
Abstract : Lie algebras are parts of the mathematical language describing symmetries. As a generalization, a motivation of studying Lie superalgebras came from supersymmetry in mathematical physics. In this talk, I will explain the basic definitions and examples of Lie superalgebras. Moreover, we will see how the structures of their root system works well as the simple Lie algebras case, especially for $\matrhfrak{gl}(m|n)$, $\mathfrak{spo}(2m|2n+1)$, and $\mathfrak{spo}(2m, 2n)$. This talk is based on Chapter 1 of the book Cheng and Wang.

Speaker : 우루노 아끼또 (Seoul National University)

Title : Crystal base of the negative half of the quantum superalgebra Uq(gl(m|n))  
Abstract : We construct a crystal base of Uq(gl(m|n))−, the negative half of the quantum superalgebra Uq(gl(m|n)). We give a combinatorial description of the associated crystal Bm|n(∞), which is equal to the limit of the crystals of the (q-deformed) Kac modules K(λ). 

Speaker : 권재훈 (Seoul National University)

Title : Introduction to super duality

Abstract : The theory of super duality is about an equivalence between two parabolic analogs of BGG categories for classical Lie algebras and Lie superalgebras of infinite ranks introduced by Cheng-Lam for type A and Cheng-Lam-Wang for orthosymplectic cases. As an application, it provides a solution to the irreducible characters of general linear and orthosymplectic Lie superalgebras. This lecture briefly reviews the theory of super duality and related works.

Speaker : 이신명 (Seoul National University)

Title : When a rep theorist met math physics
Abstract : The representation theory of Lie algebras and quantum groups has a number of connections with mathematical physics, which are not only beautiful but also have proved to be greatly beneficial to each other. 

In this talk, we illustrate one of such interplays in the spectral problem of Hamiltonians of two closely related quantum integrable models.

The first example is a Gaudin model, obtained by taking q →1 in an XXZ spin chain and so realized by a tensor product of evaluation representations of an affine Lie algebra. Finding Hamiltonians in the center of the universal enveloping algebra at the critical level (a.k.a. classical W-algebra), we construct common eigenstates by restricting invariant functionals on a tensor product of Wakimoto modules, where Bethe ansatz equations appear as analyticity conditions on their eigenvalues. 

Next, we go back to a (general) XXZ spin chain. Thanks to the universal R-matrix, this time one can directly relate representations of quantum affine algebras to central elements (including Hamiltonians), so that we may utilize known structures of the Grothendieck ring such as Baxter's TQ relations. Consequently, it enables us to compute eigenvalues in terms of q-characters without constructing eigenvectors, where Bethe ansatz equations arise in a similar manner as above.

Speaker : 윤상원 (Seoul National University)

Title : Construction of supersymmetry in vertex algebras
Abstract : We can consider the Lie superalgebras as a supersymmetric counterpart of the Lie algebra, and in the case of vertex algebras, the supersymmetric version (SUSY vertex algebra) was introduced by R. Heluani and V. G. Kac.

In this talk, first of all, by investigating the relationship between the vertex algebras and supersymmetric vertex algebras, we introduce some conditions when a given vertex algebra can be extended to a supersymmetric vertex algebra.

Based on that observation, we can also discuss 'superalgebras', particularly Lie superalgebras, superconformal algebras, and supersymmetric vertex algebras, within a unified framework.

If time permits, we also deal with some realization of supersymmetries by finding hidden superconformal structures in vertex algebras. This talk will be based on a part of the joint work with Uhi Rinn Suh. 

Speaker : 송아림 (Seoul National University)

Title : Representation Theory of Lie Superalgebras

 Abstract : Lie superalgebras are the generalization of Lie algebras with $\mathbb{Z}/2\mathbb{Z}$-gradation. The structure of Lie superalgebras gives a uniform way to explain the action of bosonic and fermionic particles, resulting in a strong motivation to study their representation theory. In this talk, I will explain the well-known facts about the finite-dimensional modules of Lie superalgebras, including the classification of simple modules, central characters, etc. For simplicity, I will mainly focus on the $\mathfrak{gl}(m|n)$ case. This talk is based on Chapter 2 of the book by Cheng and Wang.

Speaker : 장일승 (Incheon National University)

Title :  On quantum virtual Grothendieck rings
Abstract : Let C(q,t) be the two-parameter deformation of a Cartan matrix C introduced by Frenkel-Reshetikhin. Then C(t) := C(1,t) is called the t-quantized Cartan matrix of C. A quantum virtual Grothendieck ring is defined by an intersection of the kernel of deformed screening operators associated with a t-quantized Cartan matrix C(t). When C is of simply-laced type, it recovers the quantum Grothendieck ring for a category of finite-dimensional modules over untwisted quantum affine algebra of C. In this talk, I will explain and discuss the structure of quantum virtual Grothendieck rings from the viewpoint of its relationship with dual canonical basis and quantum cluster algebra structure. This talk is based on joint works with Kyu-Hwan Lee and Se-jin Oh.

Speaker : 이수홍 (Seoul National University)

Title : Categorifying Lie superalgebras
Abstract : The introduction of diagrammatic algebras categorifying the quantized enveloping algebras of Kac-Moody Lie algebras has been successful and has produced numerous striking applications. An analogous result for Lie superalgebras remains elusive, but there have been several approaches that have led to the categorification of a certain subclass that does not deviate too much from the non-super case. In this talk, I will explain these results. A main difference compared to the non-super case is the use of differential graded algebras and their derived categories.

Speaker : 이승진 (Seoul National University)

Title : Lusztig's q-weight multiplicity for type C and semistandard oscillating tableaux
Abstract : Lusztig's q-weight multiplicity is a q-analogue of weight multiplicity, which is the number of semistandard young tableaux of given shape and weight for type A. A statistic called charge on semistandard young tableaux describes the q-weight multiplicity for type A. In this talk, we discuss type C version of this theory by investigating an energy function, which appears in affine crystal theory, on semistandard oscillating tableaux.

Speaker : 이소연 (Sogang University)

Title : Regular Schur labeled skew shape posets and their 0-Hecke modules
Abstract :  Let us consider two important families of labeled posets: regular labeled posets and Schur labeled skew shape posets. The former was introduced by Björner and Wachs to characterize the conditions under which the set of  linear extensions of a labeled poset forms a weak Bruhat interval. The latter appeared in Stanley's conjecture, which states that the generating function $K_{(P, \omega)}$ of a labeled poset $(P, \omega)$ is symmetric if and only if it is isomorphic to a Schur labeled skew shape poset.

 In 1993, Duchamp, Hivert, and Thibon established a connection between $0$-Hecke modules and the generating functions of labeled posets. They constructed the $0$-Hecke modules $M_{(P, \omega)}$ associated with labeled posets $(P, \omega)$, where the quasisymmetric characteristic image of the module is the generating function $K_{(P, \omega)}$. In this talk, we specifically study $0$-Hecke modules associated with regular Schur labeled skew shape posets. We classify these modules up to isomorphism and investigate their structural properties. We also provide a combinatorial characterization of regular Schur labeled skew shape posets.

 This work is joint with Young-Hun Kim and Young-Tak Oh.

Speaker : 김소연 (University of California, Davis)

Title : Cluster algebra structures of open positroid varieties
Abstract : In this talk, I will explain an explicit combinatorial description of cluster algebra structures in the homogeneous coordinate ring of open positroid varieties $\Pi_{\mathcal{M}}^{\circ}$. This cluster algebra structure is encoded in the quiver of a plabic graph arising from a given positroid $\mathcal{M}$, which was first made explicit by the work of Muller-Speyer and proved by Lam-Galashin. Interestingly, these cluster algebras are locally acyclic, which satisfy many interesting properties. I will briefly mention the significance of this cluster algebra structure to some of the combinatorics associated with cohomological structure.