召集人：张红莲（上海大学）、VYACHESLAV FUTORNY（Southern University of Science and Technology）

时   间：2024.06.02—2024.06.08

Speakers of Mini-courses for June 2-5

1. Tomoyuki Arakawa   Ningbo University & Kyoto University  

2. Haisheng Li          Rutgers University-Camden

3. Yucai Su             Jimei University & Tongji University

4. Kaiming Zhao        Wilfrid Laurier University

Schedule of Mini-courses (June 2-5)

Abstracts: Mini-courses (June 2-5)

Tomoyuki Arakawa

 Ningbo University & Kyoto University  

Title：Symplectic singularities and vertex algebras

Abstract: Symplectic singularities introduced by Beauville appear in various aspects of representation theory and are referred to as "21st-century Lie theory" by Okounkov. On the other hand, symplectic singularities also arise in the context of quantum field theory in physics, particularly in the Higgs and Coulomb branches of three-dimensional theories, as well as in the Higgs branches of four-dimensional theories. For example, the Higgs branch of 4-dimensional Yang-Mills theory with N=4 supersymmetry is naturally defined as a quotient of symplectic singularities with respect to the Weyl group of the gauge group G.

Additionally, in vertex algebra theory, certain Poisson varieties called associated varieties are defined as geometric invariants, and they often turn out to be symplectic singularities. In such cases, vertex algebras can be regarded as chiral noncommutative deformations of symplectic singularities.

In particular, the 4D/2D duality proposed by Beem et al. in theoretical physics determines vertex algebras as invariants for superconformal four-dimensional theories such as Yang-Mills theory with N=4 supersymmetry. It is claimed that the Higgs branch of four-dimensional theories can be reconstructed as the associated variety of vertex algebras. Therefore, all vertex algebras arising from four-dimensional theories are supposed to be chiral noncommutative deformations of singular symplectic varieties.

In this lecture, we will discuss such vertex algebras and their representation theory.

Haisheng Li          

Rutgers University-Camden

Title： Quantum vertex algebras and their modules

Abstract：The aim of this short course is to give an introduction to quantum vertex algebras, with a focus on certain selected topics. The following are the main contents:

Part I: We first begin with the definitions of a vertex algebra and a module for a vertex algebra, and then present a conceptual construction of vertex algebras and modules, and give the natural association of vertex algebras and modules to affine Lie algebras.

Part II: We shall study nonlocal vertex algebras, weak quantum vertex algebras, and a conceptual construction. We shall also study quantum vertex algebras and non-degeneracy of nonlocal vertex algebras.

Part III:  We shall study $\phi$-coordinated modules for (weak) quantum vertex algebras and present a conceptual construction.

Part IV:  We present some examples of quantum vertex algebras. If time permits, we shall also discuss $\hbar$-adic quantum vertex algebras.

Yucai Su

Jimei University & Tongji University

Title：Introduction to Representations of Classical Lie Superalgebras

Abstract: This mini-course gives a brief introduction to representations of classical Lie superalgebras, suitable for beginners. We prepare to give 4 hours lectures.

1. Definition of Lie superalgebras, Kac’s classification of simple dimensional simple Lie superalgebras.

2. Representation of gl(m|m).

3. Classification of Kac-modules of gl(m|n) in Category O.

4. Mixed cohomology groups of Lie super algebras.

Kaiming Zhao

Wilfrid Laurier University

Title：Smooth representations of Lie Algebras

Abstract: This is a mini-course for graduate students and young Lie algebra researchers. This course starts with basic knowledge on representations of Lie algebras. I will start reviewing some known results on representations of various finite and infinite-dimensional Lie algebras, including the Virasoro algebra, and Virasoro superalgebras. Then we will present various simple smooth representations of some Lie algebras and the Virasoro algebra, including classification of simple smooth modules over the Virasoro algebra, Virasoro superalgebras, and Heisenberg-Virasoro algebra. Its goal is to prepare the students for further study on the various Lie algebras and their applications.

Lecture 1. Simple Smooth representations of the Virasoro algebra

Lecture 2. Simple Smooth representations of the Ramond algebra

Lecture 3. Simple Smooth representations of the mirror Heisenberg-Virasoro algebra

Course description: This mini course will recall the structure theory and representation theory of various finite and infinite-dimensional Lie algebras, including the Virasoro algebra, and Virasoro superalgebras. Then we introduce various simple smooth representations of different Lie algebras. We will also post some open problems on various simple weight and non-weight representations of some Lie algebras. The contents of this course come from the textbook

[V. Kac, A. Raina, N. Rozhkovskaya, Bombay lectures on highest weight representations of infinite dimensional Lie algebras. Second edition. Advanced Series in Mathematical Physics, 29. World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. xii+237 pp.]

and from scattered recent research papers, including  

[1] K. Iohara, Y. Koga, Representation theory of Neveu-Schwarz and Ramond algebras: Verma modules, Adv. Math. 178 (2003) 1-65.

[2] K. Iohara, Y. Koga, Representation theory of Neveu-Schwarz and Ramond algebras: Fock modules, Ann. Inst. Fourier 53 (2003) 1755-1818.

[3] Mazorchuk, Volodymyr; Zhao, Kaiming, Simple Virasoro modules which are locally finite over a positive part. Selecta Math. (N.S.) 20 (2014), no. 3, 839–854.

[3] D. Liu, Y. Pei, L. Xia, Simple restricted modules for Neveu-Schwarz algebra, J. Agebra 546 (2020) 341-356.

[4]. H. Chen, Simple restricted modules over the Ramond algebra as weak modules for vertex operator superalgebras, J. Algebra 621 (2023) 41-57.

[5]. Y. Chen, Y. Yao, R. Shen, K. Zhao, Simple smooth modules over Ramond algebra, preprint, 2024.

Speakers of Workshop for June 5-8

1. Bingtao Cao     Yunnan University

     2. Xuanzhong Dai  Kyoto University

3. Qiang Fu        Tongji University

4. Hongyan Guo    Central China Normal University

5. Naihong Hu      East China Normal University

6. Cuibo Jiang      Shanghai Jiao Tong University

7. Iryna Kashuba    Southern University of Science and Technology

8. Libin Li          Yangzhou University

9. Dong Liu        Huzhou University

10. Luan Bezerra     Southern University of Science and Technology

11. Ke Ou           Yunnan University of Finance and Economics

12. Li Ren            Sichuan University

13. Bin Shu          East China Normal University

14. Shaobin Tan      Xiamen University

15. Lizhong Wang    Beijing University

16. Quanshui Wu     Fudan University

17. Yufeng Yao      Shanghai Maritime University

18. Nina Yu          Xiamen University

Schedule of Workshop

Schedule of Workshop for June 5-8

Abstracts: Workshop (June 5-8)

Bintao Cao (曹彬涛)

Yunnan University（云南大学）

Title：Projective modules in BGG category $\mathcal O$ for osp(3|6)

Abstract：This is a joint work with Xu LUO. We give the standard filtrations of the projective modules in integral blocks in  the BGG category $\mathcal O$ of the ortho-symplectic Lie superalgebr osp(3|6).

Xuanzhong Dai (戴煊中)

Kyoto University (京都大学)

Title：A quantization of modular forms

Abstract：In this talk, we will present a quantization of modular forms and weakly holomorphic modular forms, which recovers the Rankin-Cohen brackets—a family of bilinear operations on modular forms. It has long been speculated that the Rankin-Cohen brackets are connected to vertex operator algebras, as initially proposed by W. Eholzer, Y. Manin, and D. Zagier. Our construction naturally inherits the vertex opeartor algebra structure, with the vertex operation fully determined by a slight modification of the Rankin-Cohen brackets.

Qiang Fu (付强)

Tongji University (同济大学)

Title：On quantum affine GLN and affine quantum Schur algebras

Abstract. In this lecture, we will review some results about quantum affine gln and affine quantum Schur algebras.

Hongyan Guo (郭红艳)

(华中师范大学)

Title: Integral forms of vertex operator superalgebras and their representations

Abstract: In this talk, we introduce and study integral forms of vertex operator superalgebras. A construction of integral forms of affine vertex operator superalgebras by using Chevalley basis will be presented. We will also talk about integral forms of tensor products of rational Virasoro vertex operator algebras with the tool of binary linear codes.

Naihong Hu (胡乃红)

East China Normal University (华东师范大学)

Title：Some Progresses on Two-Parameter Quantum Groups

Abstract: In this talk, we will report 3 progresses on two-parameter quantum groups, which include (i) the existences of exotic new one-parameter quantum small groups arising from two-parameter setting; (2) the description of the Harish-Chandra theorem the generic two-parameter quantum groups; (3) RLL realization of the two-parameter quantum affine algebras of types (BCD)_n^{(1)}. This are joint work with Xiao Xu, Wang Hengyi, and Rushu Zhuang, respectively.

Libin Li（李立斌）

Yangzhou Universuty (扬州大学)

Title: The Casimir numbers of finite tensor categories

Abstract: This talk deals with the Casimir numbers for some finite rigid tensor categories with finitely many isomorphism classes of indecomposable objects, especially, for the fusion category and the representation category of Hopf algebra.

Dong Liu (刘东)

Huzhou University (湖州师范学院)

Title: Classification of simple quasifinite weight modules over the loop Neveu-Schwarz algebra

Abstract：The loop Neveu-Schwarz algebra is the Lie superalgebra of the tensor product of the N = 1 Neveu-Schwarz algebra and the Laurent polynomial algebra. In this talk we introduce some progesses on its representation theory, including the classifications of simple quasifinite weight modules, as well as the necessary and sufficient conditions for highest weight modules to be simple, to be quasifinite. This is joint with Y. Pei and Q. Wu.

Luan Bezerra

Southern University of Science and Technology(南方科技大学)

Title：Partitions with parity and representations of quantum toroidal superalgebras.

Abstract: The representation theory of quantum toroidal (super)algebras is a very technical and difficult subject. On the other hand, a large class of modules where the central element C acts by 1 have an easy description through the combinatorial framework of partitions with parity. This combinatorics is not only interesting in its own right, but it is expected to be related to other concepts such as crystal bases, BPS states, and equivariant K-theory. In this talk, I will explain how to construct these modules.

Cuibo Jiang (姜翠波)

Shanghai Jiaotong University (上海交通大学)

Title: On weights of singular vectors of universal affine vertex VOAs

Abstract: Given a universal affine vertex operator algebra, it is usually hard to determine its singular vectors in general. We will talk about our recent progress on finding weights of singular vectors. We completely determine the weights of all singular vectors with minimal conformal weights for universal affine vertex operator algebras of types A and D. For some cases, we also give the form of the singular vectors. Our method is available for all other types. If time is permitted, I will also talk about the associated varieties of simple vertex operator algebras for some cases. This talk is based on joint work with Jingtian Song.

Iryna Kashuba

Southern University of Science and Technology (南方科技大学)

Title：Generalized Imaginary Verma modules.

Abstract: We develop a general technique of constructing new irreducible weight modules for any affine Kac-Moody algebra using the parabolic induction, in the case when the Levi factor of a parabolic subalgebra is infinite-dimensional and the central charge is nonzero. Our approach uniforms and generalizes all previously known results with imposed restrictions on inducing modules.

Ke Ou (欧岢)

Yunnan University of Finance and Economics (云南财经大学)

Title：Modular (co)invariants for some modular pseudo-reflection groups

Abstract：Let K be an algebraic closed field with characteristic p>3  and X be restricted Cartan type Lie algebras W, S or H over K. The Weyl groups of X are modular pseudo-reflection groups. In this talk, we will investigate the modular (co)invariants and the absolute length of some modular pseudo-reflection groups which generalizes the Weyl groups of X. For each pseudo-reflection group G, there exists a natural representation V. The modular G-invariants of S(V) and Λ(V)⊗S(V) will be considered. Then, we will determine the block basis for the coinvariants and the absolute length of G.

Li Ren (任丽)

Sichuan University (四川大学)

Title：On Kac-Wakimoto hypothesis

Abstract：Motivated by the earlier work of Kac-Wakimoto on the coset constructions associated with affine vertex operator algebras, the categorial coset constructions are investigated and Kac-Wakimoto Hypothesis is proved under some mild conditions. In particular, the field identifications are obtained. These results are applied to the coset constructions in the theory of vertex operator algebras. This is a joint work with C.Dong and F.Xu.

Bin Shu (舒斌)

East China Normal University (华东师范大学)

Title: Birational equivalence between super and purely even Zassenhaus varieties

Abstract: In this talk, we talk about the zassenhaus varieties for basic classical Lie superalgebras in prime characteristic, and show binational equivalence between them and their purely even counterparts.  This is a joint work with Lisun Zheng and Ye Ren.

Shaobin Tan (谭绍滨)

Xiamen University (厦门大学)

Title: Vertex algebras associated with elliptic Lie algebras

Abstract：Elliptic Lie algebras of maximal type are nullity two extended affine Lie algebras, which are generalization of the affine Kac-Moody Lie algebras. It is well known that the restricted modules for any untwisted affine Kac-Moody Lie algebra are isomorphic to the modules for the associated affine vertex algebra, while the restricted modules for the twisted affine Kac-Moody Lie algebra are isomorphic to the twisted modules for the affine vertex algebra. In this talk we will recall the classification of elliptic Lie algebras of maximal type, and the notion of \Gamma vertex algebra and equivariant \phi coordinated quasi-modules for vertex algebras. I will then claim that there exist a vertex algebra V associated with any elliptic Lie algebra of maximal type and an automorphism group G of V equipped with a linear character \chi, such that the category of restricted modules for the elliptic Lie algebra is isomorphic to the category of (G, \chi)-equivariant \phi-coordinated quasi modules for the vertex algebra V. The arguments will be divided into the case for fgc type and the case for non-fgc type.

Lizhong Wang (王立中)

Peking University (北京大学)

Title: A finite group is determined by 1-2-3-\mathcal{P}-characters

Abstract：In this talk, we will show that the multiplication table of a finite group can be written out by  1-2-3-\mathcal{P}-characters or 1-2-3-super characters. As an appliction, a finite group is determined by 1-2-3-characters over any field with characteristic zero.

Quanshui Wu (吴泉水)

Fudan University (复旦大学)

Title: Module-finite algebras

Abstract: Module-finite algebras are a class of noncommutative algebras that contain a relatively large central subalgebra (that is, algebras which are finitely generated  as a module over some central subalgebra). Module-finite algebras have abundant sources, such as quantum groups/algebras at root of unity. This talk will introduce some recent research  related to module-finite algebras, including the relation between the zero sets of the discriminant ideals and irreducible representations, Poisson order, as well as noncommutative resolutions, etc. 

Yufeng Yao (姚裕丰)

Shanghai Maritime University (上海海事大学)

Title: Nilpotent orbits and representations of Lie algebras

Abstract: This is a survey of nilpotent orbits and representations in modular Lie algebras based on the work of J. C. Jantzen etc., and joint work with Bin Shu and Yunpeng Xue. We first recall the classical theory of nilpotent orbits in representations of reductive Lie algebras in prime characteristic. Then we further study nilpotent orbits of the enhanced reductive Lie algebras, and give a sufficient and necessary condition for finite nilpotent orbits. All finite nilpotent orbits are precisely parameterized, and their closure are determined.

Nina Yu (余妮娜)

Xiamen University (厦门大学)

Title: On Fusion Products of Twisted Modules in Permutation Orbifolds

Abstract: Orbifold theory examines a vertex operator algebra under the action of a finite group. The primary focus lies in understanding the representation theory for the fixed-point vertex operator subalgebra. Permutation orbifolds investigate the action of the symmetric group of degree n on the n-tensor product of a vertex operator algebra. In this talk, I will talk about our recent work on fusion products of twisted modules in permutation orbifolds. 

Participants