Shenzhen Thematic Program "Representation Theory"
February-June 2025
Scientific Committee
Efim Zelmanov
SICM-SUSTech
Efim Zelmanov
SICM-SUSTech
Jiping Zhang
Peking University
Jiping Zhang
Peking University
Vyacheslav Futorny
SICM-SUSTech
Vyacheslav Futorny
SICM-SUSTech
Nikolai Reshetikhin
BIMSA-Tsinghua
Nikolai Reshetikhin
BIMSA-Tsinghua
Vera Serganova
University of California, Berkeley
Vera Serganova
University of California, Berkeley
Tomoyuki Arakawa
University of Kyoto, Japan
Tomoyuki Arakawa
University of Kyoto, Japan
Noncommutative Algebra
2025-03-17 - 2025-03-24
Speaker: Joao
Venue: 240A
\textbf{Lecture one - Growth of sequences, functions, and the GK-dimension}
\text{Duration: Mon.16:00-17:30}
\text{In this lecture we are going to consider the asymptotic growth of sequences of positive integers and, more generally, functions. We will then specify the theory of growth to affine algebras, not necessarily associative, and define the Gelfand-Kirillov dimension, showing its basic properties, and in particular, we will discuss Bergman's Gap.}
\textbf{Lecture two - Growth of associated algebras and modules}
\text{Duration: Tues.16:00-17:30}
\text{In this lecture, restricted to the case of associative algebras, we will see how the GK-dim changes when we modify our algebras, such as localizing and taking tensor products. We will also discuss the Gelfand-Kirillov dimension of modules.}
\textbf{Lecture three - GK-dim, Hilbert Polynomials and Poincaré Series}
\text{Duration: Wed. 10:00-11:30}
\text{In this lecture, we will study canonical filtered/graded techniques in ring theory and apply them to the study of the GK-dim of filtered/graded algebras. We will learn how to read information about this invariant from its Poincaré Series and Hilbert Polynomials.}
\textbf{Lecture four - Algebras with multiplicity; Bernstein and Gabber's inequality}
\text{Duration: Thur. 16:00-17:30}
\text{We will discuss the notion of holonomic modules for associative algebras, and prove the two most important examples of this phenomena: the Bernstein inequality for the Weyl algebras and Gabber's inequality for algebraic Lie algebras.}
\textbf{Lecture five - Noncommutative transcendence degrees and epilogue}
\text{Duration:Fri. 16:00-17:30}
\text{In this lecture we will discuss two attempts to generalize the notion of transcendence degree to division algebras - the Gelfand-Kirillov transcendence degree and the Lower transcendence degree. The development of noncommutative 'birational' invariants became increasingly important with the understanding of the skew field of fractions of many objects in generalized Lie theory and quantum groups; and the development of noncommutative 'projective' algebraic geometry. Finally, we will briefly mention the most important results concerning the Gelfand-Kirillov dimension obtained in the last years.}
Diagram categories and invariants of Lie algebras and quantum groups
2025-03-18 - 2025-03-21
Speaker: Ruibin Zhang(University of Sydney)
Venue: 240A
\textbf{Introduction}
Duration: 14:00-15:30
Graphical methods have been widely applied in recent years to analyse certain tensor products of modules for Lie (super)algebras and quantum (super)groups. One creates diagram categories, such as the tangle category and Brauer category, and seeks functors from them to categories of modules for these algebraic structures. The graphical description of morphisms of a diagram category usually leads to simpler descriptions of morphisms in the target category of modules, and hence a better understanding of the latter.
We discuss the Brauer category and various incarnations of it, and explain how they provide the natural context for a wide class of algebras, such as the Hecke algebra, BMW algebra, (affine) Brauer algebra, the algebra of chord diagrams, and etc., which play important roles in representation theory. We apply these categories to investigate invariants in modules of the type \(M \otimes V \otimes \cdots \otimes V\) for classical Lie (super)algebras and related quantum (super)groups, where \(V\) is the natural module and \(M\) is an arbitrary module. In particular, if \(M = V\), we obtain generalisations of Schur-Weyl dualities; and if \(M\) is taken to be the universal enveloping algebra itself, we obtain explicit generators for the centre, and derive a categorical interpretation of certain ``characteristic identities'' of the orthogonal and symplectic Lie algebras.
\textbf{Topics to be covered include}
1. \textbf{Brauer category}
- Brauer category and its oriented analogue
- Categorical Schur-Weyl-Brauer dualities for classical Lie supergroups
- Triangular decomposition of Brauer category
- Deligne category
- Indecomposable objects in the category of tensor modules for OSp
- Enhanced Brauer category and invariants of orthogonal Lie algebra
2. \textbf{Polar Brauer category}
- Polar Brauer category
- Centre of universal enveloping superalgebra
- Characteristic identities for Lie algebras
- Idempotent completion of polar Brauer category
3. \textbf{Tangle categories}
- Braid group representations and quantum R-matrices
- Tangle categories – oriented and un-oriented
- Categorical Schur-Weyl-Brauer dualities for quantum (super)groups
- Invariants of quantum \(G_2\)-module algebras
4. \textbf{Polar tangle categories}
- Polar tangle categories
- Affine Temperley-Lieb category and category \(O\) of \(U_q(\mathfrak{sl}_2)\)
Vertex operator algebras via physics
2025-04-14 - 2025-04-18
Speaker: David Ridout (University of Melbourne)
Venue: 240A
Duration: 14:00-15:30（Except Wednesday）
Vertex (operator) algebras were introduced in mathematics in the mid- to late-80s. But, at this time they were already very well understood, at least through examples, by physicists who called them chiral lgebras of conformal field theories.
These lectures will try to explain, using simple examples, the physicists' approach to VOAs. The aim is to show how these algebras appear concretely and thereby try to explain how physicists discovered so many amazing connections between VOAs are other parts of mathematics.
From Weyl Algebras to Chiral Algebras
2025-05-20 - 2025-05-23
Speaker: Libor Krizka
Venue: 240A
Duration: Tue.&Wed.&Thur.14:00-16:00
Fri.15:00-17:00
This lecture series will follow a path from Weyl algebras to chiral algebras, which serve as vertex algebra analogues of differential operator algebras. We will explore various realizations of Lie algebras and affine vertex algebras through differential operators, with a geometric perspective grounded in D-modules on flag varieties and the Beilinson-Bernstein correspondence. Special emphasis will be placed on explicit constructions of Verma modules, contragredient Verma modules, and Whittaker modules, illustrating how these classical representation-theoretic objects emerge within the chiral and geometric framework. Throughout, the focus will remain on concrete examples that illuminate the algebraic and geometric structures.
Quantum Groups Schedule
March 23 to 27 (Shenzhen)/March 28 to 31 (Xiamen)
**Monday, 24/03**
- 09:00 - 09:30: Registration
- 09:30 - 10:30: Nicolai Reshetikhin (Tsinghua University, BIMSA)
- 10:30 - 11:00: Coffee break
- 11:00 - 12:00: Evgeny Mukhin (Indiana University)
- 12:00 - 14:00: Lunch Time
- 14:00 - 15:00: Yunnan Li (Guangzhou University)
- 15:00 - 15:30: Coffee break
- 15:30 - 16:30: Hao Chang (Central China Normal University)
- 16:30 - 17:30: Xingpeng Liu (SUSTech, SICM)
**Tuesday, 25/03**
- 09:00 - 10:00: Ruibin Zhang (University of Sydney)
- 10:00 - 10:30: Coffee break
- 10:30 - 11:30: Naihong Hu (East China Normal University)
- 11:30 - 12:30: Venkatesh Rajendran (Indian Institute of Science)
- 12:30 - 14:00: Lunch Time
- 14:00 - 15:00: Duncan Laurie (University of Edinburgh)
- 15:00 - 15:30: Coffee break
- 15:30 - 16:30: Xiaomeng Xu (Peking University)
- 16:30 - 17:30: Hongdi Huang (Shanghai University)
**Wednesday, 26/03**
- 09:00 - 10:00: Olivier Mathieu (CNRS)
- 10:00 - 10:30: Coffee break
- 10:30 - 11:30: Hiroyuki Yamane (University of Toyama)
- 11:30 - 12:30: Adriano Moura (UNICAMP)
- 12:30 - 14:00: Lunch Time
- 14:00 - 17:30: Discussions
Adriano Adrega de Moura
Universidade Estadual de Campinas - Associate Professor Mar.3-28
Bart Vlaar
(BIMSA, China) Mar.26-28
Dandan Yang（杨丹丹）
Xidian University Mar.24-29
Duncan Laurie
University of Edinburgh, Postdoctoral Research Associate Mar.20-27
Evgeny Mukhin
Indiana University Indianapolis Mar.15-30
Ge Feng（冯鸽）
University of Shanghai for Science and Technology & Assistant Professor Mar.24-27
Hao Chang（常浩）
Central China Normal U Mar.23-27
Hiroyuki Yamane
(Toyama University) Mar.18-27
Hongdi Huang
Shanghai University, assistant professor Mar.23-27
Honglian Zhang(张红莲)
Shanghai University Mar.25-28
Jian Zhang(张健)
华中师范大学 Mar.21-25
Libor Krizka
RSJ Mar.22-Jun.1
Naihong Hu （胡乃红）
East China Normal University Mar.23-27
Nicolai Reshetikhin
Tsinghua Mar.23-27
Olivier Mathieu
(University of Lyon, France) Feb-Jun
Qing Wang（王清）
( Xiamen University ) Mar.23-26
Ruibin Zhang（张瑞斌）
(University of Sydney, Australia) Mar.17-30
Santanu Tantubay
Postdoc SUSTech ICM
Venkatesh Rajendran
( Indian Institute of Science, Associate Professor ) Mar.23-27
Viktor Bekkert
Federal University of Minas Gerais, Belo Horizonte (UFMG), Mar.5-May.12
Xiangqian Guo（郭向前）
Guangzhou University Mar.23-27
Xiaomeng Xu (徐晓濛 )
Peking University Mar.24-25
Yunnan Li（黎允楠）
Guangzhou University Mar.23-27
Constructing Families of Strongly Real Modules
Adriano Moura (UNICAMP), aamoura@uunicamp.br
One of the most challenging problems in the realm of the category of finite-dimensional representations of a quantum affine algebras is that of understanding necessary and sufficient conditions for a tensor product of two given simple modules to be simple as well. A simple module is said to be real if it's tensor square is simple. Such modules play a prominent role in the relation of this category with the theory of cluster algebras.
We will present results based on a joint work with M. Brito and C. Silva where we introduce the notion of strongly real modules and describe methods for constructing examples of such modules. The methods are based on the notion of reality determining subgraphs of pseudo q-factorization graphs, also introduced in this work, which arises from a result describing a sufficient condition for a module to be real. Time permitting, we shall briefly discuss a result concerning the prime factorization of modules having prime snake support in terms of sequences of maximal totally ordered subgraphs of their q-factorization graph.
Drinfeld realization of UqD(1)(2, 1; α), and Typical irreducible character formula for a generalized quantum group
Hiroyuki Yamane (University of Toyama), hiroyuki@sci.u-toyama.ac.jp
: My talk will be composed of the two parts: (1) I explain the typical irreducible character formula for a generalized quantum group U (J. Algebra Appl. 20 (2021), no. 1, 2140014). U can be the quantum algebra Uqg of a basic classical Lie superalgebra g. If U = Uqg, there are two finite dimensional irreducible highest weight U modules V (λ) and V (µ) such that V (λ) is typical and V (µ) is not typical, their highest weights λ and µ looks very similar, but their characters are really different (Appendix of arXiv:1909.08881). (2) Although it is an old topic, I will explain Drinfeld's second realization of UqD(1)(2, 1; α) (RIMS Kokyuroku Bessatsu B8 (2008), 171-216). This part is a joint work with I. Heckenberger, F. Spill, and A. Torrielli.
Free Jordan Algebras and DGA
Olivier Mathieu (CNRS, Lyon), mathieu@math.univ-lyon1.fr
The free Jordan algebra J(2) over 2-generators has been explicitly de-
scribed by Shirshov in his 1956 paper. For D ≥ 3 the free Jordan algebras J(D) remain an elusive object.
In a joint paper with I. Kashuba, we introduce an explicit conjecture for
the dimension of the degree n component Jn(D) and in joint preprint with
M. Lau, this conjecture has been interpreted in terms of Ext-groups in a
category of representations of the Lie algebra ˆsl2(J), the universal central
extension of the TKK-algebra.
Some initial approaches using representations of quantum groups or other
monoidal categories seem now hopeless. Thus in a work in progress with V.
Dotsenko, we introduce the notion of DGA in the Tits category. Basically it
means that a failure of the conjecture should be connected with the existence
of additional k-ary operations, k > 2.
Matched pairs, braiding operators and the Yang-Baxter equation
Yunnan Li (Guangzhou University), ynli@gzhu.edu.cn
An effective and fruitful approach to study set-theoretic solutions of the Yang-Baxter equation is to identify and investigate the underlying algebraic structures. In this talk, I will introduce several equivalent algebraic tools providing set-theoretic type solutions to the Yang-Baxter equation, namely matched pair and braiding operator, etc, constructed originally on groups and further on Hopf algebras, and then discuss some pertinent problems about them.
Modular representations of the Yangian Y_2
Hao Chang (Central China Normal University), xxu@bicmr.pku.edu.cn
Let Y_2 be the Yangian associated to the general linear Lie algebra gl_2 defined over an algebraically closed
field k of characteristic p>0. In this talk, I will discuss recent progress on the the representation theory of the restricted Yangian Y_2^{[p]}.
This leads to a description of the representations of gl_{2n}, whose p-character is nilpotent with Jordan type given by a two-row partition (n, n).
This is joint work with Jinxin Hu and Lewis Topley.
NON-FLAT BRAIDED SYMMETRIC ALGEBRAS AND QUANTUM G2 INVARIANTS
Ruibin Zhang (University of Sydney), ruibin.zhang@sydney.edu.au
We study a class of module algebras over Uq = Uq(G2), which are non-flat deformations of symmetric algebras. Let V = C(q) 7 be the 7-dimensional simple Uqmodule, and let Sq(V ) be its quantum symmetric algebra. The braided tensor product algebras Am = Sq(V ) ⊗m (with braided multiplications) are non-flat deformations of S(C 7 ⊗ C m) for all m ≥ 1. (1) We decompose Sq(V ) at generic q into a direct sum of simple Uq-submodules, and determine their multiplicities. Using this result, we construct the affine schemes associated with the classical limits of Am for all m. (2) We construct a finite set of explicit generators for the subalgebra A Uq m of Uqinvariants in Am, and determine the commutation relations among the generators. These results in particular enable one to describe the subspace of invariants of any tensor power of V , thus may be regarded as a noncommutative analogue of the first fundamental theorem of invariant theory for Uq. This is joint work with Hongmei Hu.
On multiplicity-free weight modules over quantum affine algebras
Xingpeng Liu (SICM SUSTech), liuxp@sustech.edu.cn
Multiplicity-free weight modules are a special class of infinite-dimensional weight modules characterized by weight multiplicities uniformly bounded by 1. These modules not only play a crucial role in the classification of simple weight modules but also serve as key examples in understanding the existence and construction of bounded weight modules. In this talk, I will discuss an effective method for constructing multiplicity-free weight modules for quantum affine algebras, which provide deeper insights into the underlying algebraic structures. This work was joint with V. Futorny.
Quantum supersymmetries and two quantum de Rham super complexes and stratification Theorem of indecomposable modules
Naihong Hu (East China Normal University)., nhhu@math.ecnu.edu.cn
In order to study the ``modular" representation theory of quantum gl(m|n) at root of unity, we introduce the quantum Manin superspace and quantum (dual) Grassmann superalgebra with quantum divided power structure, and develop a kind of quantum differential calculus over them, and construct two kinds of quantum de Rham super complexes: one is of infinite length which is the quantized version of the classical analogue due to Manin-Deligne-Morgan in their early study of supermanifolds from gauge field theory, another is of finite length which has no classical analogue to our knowledge. For the latter, we prove the Poincare lemma for non-truncated complexes, while for the truncated case, in order to calculate all the quantum de Rham cohomologies we need to develop a specific technique to overcome the complicated difficulties encountered in the quantum supercase. I'll also talk about the ``$\ell$-adic phenomenon" occurred in a kind of indecomposable modules in the root of unity case which originally were irreducible modules in the generic case. This talk is based on a series of our joint work with Dr. Ge Feng, and Prof. Marc Rosso.
Recent advances for universal K-matrices
Bart Vlaar (BIMSA) (For Xiamen), b.vlaar@bimsa.cn
R-matrices, solutions of cubic braid relations, arise naturally in representations of quasitriangular bialgebras. One can also consider (tensor) K-matrices, solutions of cylindrical (quartic) braid relations. They appear in a more refined algebraic context, involving a pair consisting of a quasitriangular bialgebra A and coideal subalgebra B. The key family of examples is given by quantum symmetric pairs where A is a Drinfeld-Jimbo quantum group and B a Letzter-Kolb q-deformed fixed-point subalgebra (also known as i-quantum group).
I will review recent joint work with Andrea Appel. In particular, we will see that tensor K-matrices canonically yield ring homomorphisms from the Grothendieck ring of finite-dimensional A-modules to (a commutative subalgebra of) B. If A is a quantum group of affine type then the construction is valid upon extending scalars to formal series. It specializes in existing constructions in quantum integrable systems with boundaries due to Sklyanin. Also, I outline possible future work in the direction of universal descriptions of spectra of such systems (in the style of E. Frenkel and D. Hernandez) and "boundary analogues" of the Frenkel-Mukhin-Reshetikhin q-characters.
Representations of quantum affine sl(2).
Evgeny Mukhin (Indiana University, Indianapolis), emukhin@iu.edu
I will discuss the representations of quantum affine sl(2) at generic values of q.
I will formulate a (large) number of questions and provide a (small) number of answers.
This talk is based on a joint work with A. Grigorev.
Root generated subalgebras of symmetrizable Kac-Moody algebras
Venkatesh Rajendran (Indian Institute of Science), rvenkat@iisc.ac.in
The derived algebra of a symmetrizable Kac-Moody algebra G is generated, as a Lie algebra, by the root spaces associated with its real roots. In this talk, we explore the natural reverse question: given any subset of real root vectors of G, is the Lie subalgebra of G they generate also the derived algebra of a Kac-Moody algebra? We refer to such Lie subalgebras as root-generated and provide an affirmative answer to this question. Furthermore, we establish a one-to-one correspondence between these subalgebras, real closed subroot systems, and π-systems contained within the positive system of G. This is joint work with Deniz Kus and Irfan Habib.
Stokes phenomenon and WKB approximation in representation theory
Xiaomeng Xu (Peking University) , xxu@bicmr.pku.edu.cn
This talk gives an introduction to the Stokes phenomenon and WKB analysis of confluent hypergeometric differential equations. It then realizes various structures in the representation theory via the Stokes phenomenon, like Yangians, crystal structures of quantum groups, Gelfand-Tsetlin basis, Robinson–Schensted correspondence, Littlewood–Richardson rule, Schutzenberger involution and so on.
Tensor products and R-matrices for quantum toroidal algebras
Duncan Laurie (The University of Edinburgh), duncan.laurie@ed.ac.uk
Quantum toroidal algebras are the ‘double affine’ objects within the quantum setting. In particular, they contain – and are generated by – horizontal and vertical quantum affine subalgebras Uh and Uv.
These quantum toroidal algebras are not known to possess a coproduct, but do have topological coproducts Δu mapping to a completion of the tensor square. However, this still fails to produce a tensor product on the category of integrable modules due to the presence of infinite sums.
Our solution is to conjugate Δu with an anti-involution that swaps Uh and Uv. All of our problems then fall away, and we obtain well-defined tensor products satisfying various nice properties, as well as toroidal R-matrices that solve the Yang-Baxter equation.
Twisting Manin's universal quantum groups and comodule algebras
Hondi Huang (Shanghai University), hdhuang@shu.edu.cn
In this talk, we introduce the notion of quantum-symmetric equivalence of two connected graded algebras, based on Morita--Takeuchi equivalences of their universal quantum groups. We will discuss the algebraic invariants of quantum-symmetric equivalence classes and its application. By combining our results with the work of Raedschelders and Van den Bergh, we prove that Koszul Artin--Schelter regular algebras of a fixed global dimension form a single quantum-symmetric equivalence class.
Vertex algebras Schedule
April 20-26, 2025
21/04 MONDAY
9:00 - 9:30: Registration
9:30 - 10:30: Tomoyuki Arakawa (University of Kyoto)
10:30 - 11:00: ✔ Coffee break
11:00 - 12:00: Qing Wang (Xiamen University)
12:00 - 14:00: ✔ Lunchtime
14:00 - 15:00: Shigenori Nakatsuka (University of Alberta)
15:00 - 15:30: ✔ Coffee break
15:30 - 16:30: Justine Fasquel (University of Melbourne)
22/04 TUESDAY
9:30 - 10:30: Naihuan Jing (North Carolina State University)
10:30 - 11:00: ✔ Coffee break
11:00 - 12:00: Jinwei Yang (Shanghai Jiaotong University)
12:00 - 14:00: ✔ Lunchtime
14:00 - 15:00: Xuanzhong Dai (RIMS, Kyoto University)
15:00 - 15:30: ✔ Coffee break
15:30 - 16:30: Discussions
23/04 WEDNESDAY
9:30 - 10:30: Simon Wood (Cardiff University)
10:30 - 11:00: ✔ Coffee break
11:00 - 12:00: Libor Krizka (Charles University)
12:00 - 14:00: ✔ Lunchtime
15:30 - 16:00: Discussions
16:00 - 16:30: ✔ Coffee break
24/04 THURSDAY
9:30 - 10:30: David Ridout (University of Melbourne)
10:30 - 11:00: ✔ Coffee break
11:00 - 12:00: Christopher Raymond (Hamburg University)
12:00 - 14:00: ✔ Lunchtime
14:00 - 15:00: Hao Zhang (Tsinghua University)
15:00 - 15:30: ✔ Coffee break
15:30 - 16:30: Discussions
25/04 FRIDAY
9:30 - 10:30: Drazen Adamovic (University of Zagreb)
10:30 - 11:00: ✔ Coffee break
11:00 - 12:00: Yevhen Makedonskyi (BIMSA)
12:00 - 14:00: ✔ Lunchtime
14:00 - 15:00: Discussions
15:00 - 15:30: ✔ Coffee break
15:30 - 16:30: Discussions
Christopher Raymond
University of Hamburg
Cuibo Jiang
Shanghai Jiaotong University
David Ridout
University of Melbourne
Drazen Adamovich
University of Zagreb
Hao Zhang
清华大学
Jinwei Yang
Shanghai Jiaotong University
Justine Fasquel
University of Melbourne
LI MING(李明)
TYUT
Libor Krizka
RSJ
Naihuan Jing
North Carolina State University
Qing Wang
Xiamen University
Shigenori Nakatsuka
University of Alberta
Simon Wood
Cardiff University
Tomoyuki Arakawa
University of Kyoto
Xuanzhong Dai
RIMS, Kyoto University
Yevgen Makedonski
BIMSA
Associated varieties of simple affine vertex algebras at non-admissible levels
Libor Krizka (Charles University), krizka.libor@gmail.com
In this talk, we introduce a method for determining the associated variety of a simple affine vertex algebra. We use it for describing the associated variety of the simple affine vertex algebra $\mathcal{V}_k(\mathfrak{sl}_{r+1})$ at the boundary non-admissible level $k+r+1=\frac{r}{q}$ with $\gcd(q,r)=1$. Moreover, we also get a free field realization of the simple affine vertex algebra $\mathcal{V}_k(\mathfrak{sl}_{r+1})$ and the simple affine $\mathcal{W}$-algebra $\mathcal{W}_k(\mathfrak{sl}_{r+1},f_{\rm min})$ at that level.
Affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(l,0)$ at admissible level
Qing Wang（Xiamen University）, https://math.xmu.edu.cn/info/1081/7714.htm
We present our recent results on affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(l,0)$ at admissible level $l$. We prove that the category of weak $L_{\widehat{osp(1|2)}}(l,0)$-modules on which the positive part of $\widehat{osp(1|2)}$ acts locally nilpotently is semisimple. Then we prove that Q-graded vertex operator superalgebras $(L_{\widehat{osp(1|2)}}(l,0),\omega_\xi)$ with a new Virasoro element $\omega_\xi$ are rational and the irreducible modules are exactly the admissible modules for $\widehat{osp(1|2)}$, where $0<\xi<1$ is a rational number. Furthermore, we determine the Zhu's algebras $A(L_{\widehat{osp(1|2)}}(l,0))$ and their bimodules $A(L(l,j))$ for $(L_{\widehat{osp(1|2)}}(l,0),\omega_\xi)$, where $j$ is the admissible weight. As an application, we calculate the fusion rules among the irreducible ordinary modules of $(L_{\widehat{osp(1|2)}}(l,0),\omega_\xi)$. This is a joint work with Huaimin Li.
Chiral differential operators on the base affine space and the Gelfand-Graev action
Xuanzhong Dai（RIMS, Kyoto University ）, https://www.kurims.kyoto-u.ac.jp/~xzdai/
In this talk, we investigate the global sections of chiral differential operators on the base affine space G/U, where G is a complex connected semisimple group of ADE type, and U is its maximal unipotent subgroup. We analyze the associated variety of these global sections, identifying it with the affine closure of the cotangent bundle of G/U. Furthermore, we establish that the Gelfand-Graev action naturally extends to the setting of chiral differential operators. This is joint work in progress with Tomoyuki Arakawa and Bailin Song.
Conformal blocks and coends
Hao Zhang(Tsinghua University), https://zhanghao1999math.github.io/homepage/
Conformal blocks are central objects in conformal field theories. In this talk, I will present the Sewing-Factorization (SF) theorem, a powerful result about the conformal blocks of C2-cofinite vertex operator algebras. Additionally, I will explore the connection between various coends and the SF theorem. This is based on an ongoing project with Bin Gui, initiated in arXiv:2305.10180, arXiv:2411.07707, arXiv: 2503.23995.
Duality of hook-type W-superalgebras
Shigenori Nakatsuka (University of Alberta), shigenori.nakatsuka.2022@gmail.com
W-(super)algebras are a large class of vertex algebras obtained from affine Lie algebras. Among them, the so-called hook-type W-(super)algebras play a prominent role in the picture of the webs of W-algebras.
I will explain the duality appearing at the level of algebras themselves and how it upgrades to the representation category. The first half is based on joint work with Thomas Creutzig, Andrew Linshaw, and Ryo Sato.
Duality theorems for staircase matrices.
Yevhen Makedonskyi (BIMSA), mak@bimsa.cn
The well known Cauchy identity expresses the product of terms $(1 - x_i y_j)^{-1}$ for $(i,j)$ indexing entries of a rectangular $m\times n$-matrix as a sum over partitions $\lambda$ of products of Schur polynomials: $s_{\lambda}(x)s_{\lambda}(y)$.
Algebraically, this identity comes from the decomposition of the symmetric algebra of the space of rectangular matrices, considered as a $\mathfrak{gl}_m$-$\mathfrak{gl}_n$-bi-module.
I will talk about the generalization of such decompositions by replacing rectangular matrices with arbitrary staircase-shaped matrices equipped with the left and right actions of the Borel upper-triangular subalgebras.
For any given staircase shape $\mathsf{Y}$
we describe left and right "standard" filtrations on the
symmetric algebra of the space of shape $\mathsf{Y}$ matrices. We show that the subquotients of these filtrations are tensor products of Demazure and opposite van der Kallen modules over the Borel subalgebras.
On the level of characters,
we derive three distinct expansions for the product $(1 - x_i y_j)^{-1}$ for $(i,j) \in \mathsf{Y}$. The first two expansions are sums of products of key polynomials $\kappa_\lambda(x)$ and (opposite) Demazure atoms $a^{\mu}(y)$. The third expansion is an alternating sum of products of key polynomials $\kappa_{\lambda}(x)\,\kappa^{\mu}(y)$.
The talk will be based on two papers, one of them is joint with Anton Khoroshkin and Evgeny Feigin and the second is joint with Anton Khoroshkin.
On quasi-lisse vertex algebras related to regular representations for $SL(2)$ and affine $W$--algebras
Dražen Adamović （University of Zagreb), drazen.adamovic@math.hr
We construct a family of potentially quasi-lisse (non-rational) vertex algebras, denoted by $\mathcal C_p$, $p \ge 2$, which are closely related to the vertex algebra of chiral differential operators on $SL(2)$ at level $k=-2+1/p$. We prove that for $p=3$, there is an isomorphism between $\mathcal C_3$ and the affine vertex algebra $L_{-5/3}
(\mathcal g_2)$ from Deligne's series. We establish isomorphisms between $C_p$ and certain affine $W$--algebras of types $F_4$ and $E_8$, which solve the problems of decemopostions of certain conformal embeddings. We also discuss higher rank generalization of our construction.
This talk is based on joint work with A. Milas
Quantum Affine Algebras and Twisted Extended Quantum Affine Algebras.
Naihuan Jing (North Carolina State University）, nathanjing@hotmail.com
Quantum affine algebras are quantum enveloping algebras of
affine Lie algebras, introduced independently by Drinfeld and Jimbo in
their study of the Yang-Baxter equation. Representation theory of
quantum affine algebras depend mostly on the Drinfeld realization. Quantum toroidal algebras and
Quantum extended affine algebras are
quantization of toroidal Lie algebras and extended affine algebras respectively.
We will discuss our recent joint work with F. Chen, F. Kong and S. Tan on Drinfeld realization for twisted quantum extended affine algebras, in particular we will study deformation of all nullity 2 extended affine Lie algebras.
Representation and tensor category of affine sl_2 at positive rational levels
Jinwei Yang (Shanghai Jiaotong University), jinwei2@sjtu.edu.cn
In this talk, I will discuss the tensor category structure on the categories of affine sl_2 Lie algebra modules.
The principal reduction associated to $\mathfrak{psl}_{2\vert2}$
David Ridout (University of Melbourne), david.ridout@unimelb.edu.au
The affine vertex operator superalgebra
$V^k(\mathfrak{psl}_{2\vert2})$ has many applications in physics,
including the AdS/CFT correspondence and higher-spin gravity. Its
minimal reduction, the (small) $N=4$ superconformal algebra, has
likewise been studied because of its important applications. In
contrast, its principal reduction appears to have attracted very little
attention...
Joint work with Zac Fehily and Chris Raymond
The rigidity of admissible sl2 and N=2 vertex super algebras
Simon Wood (Cardiff University), woodsi@cardiff.ac.uk
In the late 2000s and early 2010s David Ridout and Thomas Creutzig pioneered the "standard module" generalisation of the Verlinde formula and computed its predicted fusion product decompositions in the category of weight modules over affine sl2 at admissible levels. One necessary condition for these conjectured fusion rules to make sense is the rigidity of the category of weight modules. In this talk I will present recent work proving that the category of weight modules is indeed rigid and that the conjectured decompositions of fusion products of projective modules hold.
Virasoro type reductions and their inverse
Justine Fasquel (University of Melbourne), justine.fasquel@unimelb.edu.au
W-algebras form a broad family of vertex algebras built from a simple Lie algebra and parameterized by its nilpotent orbits. They are constructed by applying quantum Hamiltonian reductions to affine vertex algebras, a complex process that remains largely mysterious. Partial and inverse quantum Hamiltonian reductions have been introduced to address the intricacies of general reductions. They rely respectively on splitting and inverting the reductions.
In this talk, we will explore partial and inverse reductions of a particular type, known as the Virasoro type. Such reductions can be applied to a substantial family of W-algebras associated with height-two nilpotent orbits. Moreover, they can be lifted to the universal W-algebra of sp(2) type constructed by Creutzig, Kovalchuk and Linshaw. The talk is based on a work in collaboration with V. Kovalchuk and S. Nakatsuka.
Weight modules for sl(3) by inverse hamiltonian reduction
Christopher Raymond (University of Hamburg)christopher.raymond@uni-hamburg.de
Inverse Hamiltonian reduction has proven to be a powerful tool for understanding the representation theory of affine W-algebras. However, the state of the art still involves understanding applications of these techniques in low-rank examples. In this talk, I will introduce the ideas of inverse Hamiltonian reduction and present some recent progress on the representation theory of simple affine sl(3) VOAs at fractional admissible levels and their related W-algebras, with a focus on modules with infinite-dimensional weight spaces. This is based on joint work with David Ridout and Justine Fasquel.
Weight representations of affine Kac-Moody algebras and small quantum groups.
Tomoyuki Arakawa (University of Kyoto), arakawa@kurims.kyoto-u.ac.jp
In our previous work with Thomas Creutzig and Kazuya Kawasetsu, we conjectured that the category of weight modules over admissible affine vertex algebras is blockwise equivalent to that of the (unrolled) small quantum group, and we proved this conjecture in the case of sl_2
In this talk, we establish the conjecture for sp(2n) at the admissible level -1/2.
This is joint work with Vyacheslav Futorny, Simon Lentner, and Simon Wood.
Lie Theory Schedule
May 25-31, 2025
26/05 MONDAY
9:00 - 9:30 Registration
9:30 - 10:30 Daniel Nakano (The University of Georgia)
10:30 - 11:00 Coffee break
11:00 - 12:00 David Hernandez (Université Paris Cité)
12:00 - 14:00 Lunchtime
14:00 - 15:00 Jonathan Nilsson (Linköping University)
15:00 - 15:20 Coffee break
15:20 - 16:20 Kei Yuen Chan (The University of Hong Kong)
16:30 - 17:30 Xabier García-Martínez (Universidad de Vigo)
27/05 TUESDAY
9:00 - 10:00 Ben Webster (The University of Waterloo)
10:00 - 10:20 Coffee break
10:20 - 11:20 Kevin Coulembier (University of Sydney)
11:20 - 12:20 Xuhua He (The University of Hong Kong)
12:20 - 14:00 Lunchtime
14:00 - 15:00 Samuel Lopes (Universidade do Porto)
15:00 - 15:20 Coffee break
15:20 - 16:20 Chun-Ju Lai (Academia Sinica)
28/05 WEDNESDAY
9:00 - 10:00 Yongchang Zhu (The Hong Kong University of Science and Technology)
10:00 - 10:20 Coffee break
10:20 - 11:20 Haisheng Li (Rutgers University-Camden)
11:20 - 12:20 Arik Wilbert (The University of South Alabama)
12:20 - 14:00 Lunchtime
14:00 - 17:00 Discussions
29/05 THURSDAY
9:00 - 10:00 Olivier Mathieu (CNRS)
10:00 - 10:20 Coffee break
10:20 - 11:20 Abdenacer Makhlouf (Université de Haute-Alsace)
11:20 - 12:20 Shun-Jen Cheng (Academia Sinica)
12:20 - 14:00 Lunchtime
14:00 - 15:00 Michael Lau (Université Laval)
15:00 - 15:20 Coffee break
15:20 - 16:20 Kaveh Mousavand (The Okinawa Institute of Science and Technology)
30/05 FRIDAY
9:00 - 10:00 Kailash Misra (North Carolina State University)
10:00 - 10:20 Coffee break
10:20 - 11:20 Alistair Savage (The University of Ottawa)
11:20 - 12:20 Euiyong Park (University of Seoul)
12:20 - 14:00 Lunchtime
14:00 - 15:00 Boujemaa Agrebaoui (The University of Sfax)
15:00 - 15:20 Coffee break
15:20 - 16:20 Discussions
Abdenacer MAKHLOUF
University of Haute Alsace
Alistair Savage
University of Ottawa
Arik Wilbert
University of South Alabama
Beeri Greenfeld
University of Washington
Ben Webster
University of Waterloo
Boujemaa Agrebaoui
University of Sfax
Chengming Bai
Nankai University
Chun-Ju Lai
Academia Sinica
Daniel Nakano
University of Georgia
Dashu Xu
University of Science and Technology of China
David Hernandez
Université Paris Cité
Dong Liu
University of Science and Technology of China
Dylan Fillmore
Southern University of Science and Technology
Efim Zelmanov
Southern University of Science and Technology
Euiyong Park
University of Seoul
Farukh Mashurov
Southern University of Science and Technology
Haijun Tan
Northeast Normal University
Haisheng Li
Rutgers University-Camden
Han Dai
University of Science and Technology of China
Hongda Lin
Southern University of Science and Technology
Honglian Zhang
Shanghai University
Husileng Xiao
Harbin Engineering University
Iryna Kashuba
Southern University of Science and Technology
Jhon Alexander Ramirez Bermudez
Southern University of Science and Technology
Jinkui Wan
Beijing Institute of Technology
Joao Schwarz
Southern University of Science and Technology
Jonathan Nilsson
Linköping University
Kailash C. Misra
North Carolina State University
Kaiming Zhao
Wilfrid Laurier University
Karimbergen Kudaybergenov
Harbin Institute of Technology
Kaveh Mousavand
Okinawa Institute of Science and Technology
Kei Yuen Chan
The University of Hong Kong
Kevin Coulembier
The University of Sydney
Libor Křížka.
Charles University
Longhui Wang
University of Science and Technology of China
Luan Bezerra
Southern University of Science and Technology
Michael Lau
Université Laval
Ming Li
Taiyuan University of Technology
Minghao Bi
University of Science and Technology of China
Ning Qiao
East China Normal University
Olivier Mathieu
CNRS
Paramila Raman
Emory University
Samuel Lopes
Univerisity of Porto
Santanu Tantubay
Southern University of Science and Technology
Shaobin Tan
Xiamen University
Shun-Jen Cheng
Academia Sinica
Vyacheslav Futorny
Southern University of Science and Technology
Wooseok Jung
University of Seoul
Xabier García Martínez
Universidade de Vigo
Xiangqian Guo
Guangzhou University
Xiao He
Beijing University of Chemical Technology
Xingpeng Liu
Southern University of Science and Technology
Xuhua He
The University of Hong Kong
Yaohui Xue
Nantong University
Yongchang Zhu
Hong Kong University of Science and Technology
Yongjie Wang
Hefei University of Technology
Yucai Su
Jimei University
Yufeng Pei
Huzhou University
Affine Lie algebra weight multiplicities and
pattern avoiding permutations
Kailash Misra, misra@ncsu.edu
Affine Lie algebras form an important class of Kac-Moody Lie al_x0002_gebras. The combinatorial representation theory of affine Lie algebrashas many applications in other areas of mathematics and physics. Inthis talk we will discuss the multiplicities of maximal dominant weightsof the highest weight sl(n)- module V (kΛ0). We use the extendedYoung diagram realizations of the crystal bases of these modules toshow that these multiplicities are given by certain pattern avoidingpermutations. This talk is based on some joint work with RebeccaJayne.
Category O for Lie superalgebras
Daniel Nakano, nakano@uga.edu
In this talk, the speaker will define a general Category O for anyquasi-simple Lie superalgebra. Our construction encompasses (i) theparabolic Category O for complex semisimple Lie algebras, and (ii) theknown constructions of Category O for specific examples of classicalLie superalgebras. In particular, the speaker will develop a parabolicCategory O for classical Lie superalgebras.Connections between the categorical cohomology and the relativeLie superalgebra cohomology will be firmly established. These resultswill be used to show that the Category O is standardly stratified. Thedefinition of standardly stratified used in this context is generalizedfrom original definition of Cline, Parshall, and Scott. An explicitdescription of the categorical cohomology ring will be given, and finitegeneration results will be presented.Furthermore, it will be shown that the complexity of modules inCategory O is finite with an explicit upper bound given by the di_x0002_mension of the subspace of the odd degree elements in the given Liesuperalgebra. This result provides a generalization of prior work forgl(m|n) due to Coulembier and Serganova.This talk represents joint work with Chun-Ju Lai and Arik Wilbert.
Cluster algebras and shifted quantum groups
David Hernandez, david.hernandez@imj-prg.fr
We report on a joint work and an ongoing project with C. Geissand B. Leclerc. Shifted quantum affine algebras emerged from thestudy of quantized Coulomb branches. We show that the Grothendieckring of the category O for the shifted quantum affine algebras hasthe structure of a cluster algebra. The cluster variables of a class ofdistinguished initial seeds are certain formal power series defined froma new Weyl group action introduced in a joint work with Frenkel.These cluster variables satisfy a system of functional relations calledQQ-system. In a work in progress, we extend the construction to nonsimply-laced types, and we give a new interpretation of F-polynomialsin this context
Cocenters of Hecke algebras/categories
Xuhua He, xuhuahe@maths.hku.hk
It is known that the number of conjugacy classes of a finite groupequals the number of irreducible representations (over complex num_x0002_bers). The conjugacy classes of a finite group give a natural basisof the cocenter of its group algebra. Thus the above equality can bereformulated as a duality between the cocenter of the group algebraand the Grothendieck group of its finite dimensional representations.The situation becomes more complicated, yet more interesting, ifthe group algebras are replaced by the Hecke algebras/Hecke cate_x0002_gories. In this talk, I will explore some aspects of the theory of co_x0002_centers, with applications to arithmetic geometry and representationtheory. It is based on some of my work, and with my recent joint workwith Heleodoro and Xinwen Zhu.
Correlation Functions of Beta-Gamma and b-c
Systems on Modular Curves
Yongchang Zhu, mazhu@ust.hk
For a finite-dimensional representation of the modular group, wedefine the corresponding Beta-Gamma system and b-c system on mod_x0002_ular curves. We compute the correlation functions of these systems onmodular curves and discuss the connections between the correlationfunctions and the universal optimality of the E8 and Leech lattices.
Crystals and twist automorphisms
Euiyong Park, epark@uos.ac.kr
The (quantum) twist automorphisms on (quantum) unipotent cellstake an important position in studying quantum groups, (quantum)cluster algebras and categorification. In this talk, we will talk aboutrecent results on a new crystal-theoretic description of the quantumtwist automorphism in terms of the crystal basis theory, and showseveral interesting periodicities of twist automorphisms obtained byusing the new crystal-theoretic description. This is a join work withWoo-Seok Jung.
Cuspidal Representations over Superconformal
Algebras
Olivier Mathieu, CNRS Lyon
The Virasoro algebra Vir, which is the universal central extension of cthe Witt algebra W := Der C[t, t−1] appeared around 1960 in the worksof I. Gelfand and R. Block. In 1971, a search for superextensions ofVir started in two papers, one by A. Neveu and J. H. Schwarz and canother one by P. Ramond, followed by the works of M. Ademollo andal., K. Schoutens, A. Schwimmer and N. Seiberg.Then V. Kac and J. van de Leur have formally defined a supercon_x0002_formal algebra as a simple Z-graded Lie superalgebra L of growth onewhich contains the Witt algebra W. Together with S. Cheng, theyconjectured the full classification of all superconformal algebras.I will explain the explicit classification of all cuspidal modules overall known supercuspidal algebras of rank ≥ 1, and their central exten_x0002_sions. Most of our work is based general ideas involving the TKK_x0002_construction, locality and semi-locality. However, my report will focus on the algebras K(4), K(2)(4) andCK(6), which are special because their local, semi-local or TKK_x0002_structures present defects. As a consequence, these algebras haveexceptional representations.
Discrete series and Springer correspondence for
type H4
Kei Yuen Chan, kychan1@hku.hk
The classical Springer correspondence constructs Weyl group rep_x0002_resentations from the geometry of Springer fibers. Lusztig models thistheory in the framework of the graded Hecke algebra. From this al_x0002_gebraic viewpoint, one can establish the Springer correspondence forall finite reflection groups. For example, nilpotent orbits can be re_x0002_placed by certain combinatorial data studied by Heckman-Opdam. Inthis talk, I shall explain several ingredients to establish the Springercorrespondence for the reflection group of type H4, including the Ext_x0002_vanishing properties of discrete series. The main results in this talkare joint with Simeng Huang (Fudan University).
Endomorphism algebra of modules and their
modern applications
Kaveh Mousavand, mousavand.kaveh@gmail.com
Endomorphism algebra of modules and their modern applicationsAbstract: Let A be a finite-dimensional associative k-algebra. In clas_x0002_sical representation theory, the endomorphism algebras of A-modulesplay a central role in understanding the structure and properties ofA. A module X over A is called a brick (or a Schur representation) ifits endomorphism algebra is a division ring. Bricks have proven to becrucial in both classical and modern developments within representa_x0002_tion theory and related areas. However, as we will show in this talk,a comprehensive understanding of the behavior of bricks over a givenalgebra A remains elusive. Motivated by a series of open conjecturesposed over the past seven years, I will present some recent resultsconcerning the structure and properties of bricks. Along the way, wewill see how certain technical challenges and open problems can beapproached through the concept of Hom-orthogonal modules—an ap_x0002_proach that also opens the door to analogous problems beyond therealm of finite-dimensional associative algebras. This talk is based ona series of works, including joint projects with Charles Paquette.
Gelfand-Tsetlin modules and KLR algebras
Ben Webster, b2webste@uwaterloo.ca
I will discuss how we can use the representation theory of KLRalgebras to construct and analyze Gelfand-Tsetlin modules.
Lie algebra representations and free Jordan
algebras
Michael Lau, Michael.Lau@mat.ulaval.ca
For any unital Jordan algebra, the famous Tits-Kantor-Koecherconstruction produces an sl(2)-graded Lie algebra. We will look atweight modules for universal central extensions of these algebras, con_x0002_centrating on categories of modules satisfying combinatorial domi_x0002_nance or smoothness conditions. We describe some finiteness resultsfor algebras and Weyl modules in these contexts. Surprisingly, theproofs of several of our results use Zelmanov’s theorem on nil Jordanalgebras of bounded index. No prior knowledge of Jordan theory willbe assumed. This talk is based on joint work with Olivier Mathieu.
Lifting free modules to Generalized Weyl
Algebras
Jonathan Nilsson, jtan86@gmail.com
Generalized Weyl algebras (GWAs) are versatile objects appearingin the study of various algebraic structures including the classical Weylalgebra, quantum groups, and certain Lie algebras. The GWA R(σ, a)is generated by x,y, and R subject to commutation relations involvingan R-automorphism σ and a central element a of R. In this talk, Iwill focus on a class of modules over GWAs which are free of finiterank over the base ring R. When R is a unique factorization domainI will show how these modules can be explicitly described in termsof the divisors of the element a and the orbits of σ acting on themaximal ideals of R, and I will exhibit simple modules for each finiterank. Particular attention will be given to the rank 1 case when R isa principal ideal domain. In this setting we derive explicit criteria forsimplicity, composition series, and a description of the weight modulesappearing as simple subquotients. The talk is based on joint work withSamuel Lopes.
On the Hochschild cohomology of quantum
nilpotent algebras
Samuel Lopes, slopes@fc.up.pt
Quantum nilpotent algebras (QNA) try to replicate the structure ofenveloping algebras of nilpotent Lie algebras and their deformations,which often exhibit behaviour more akin to enveloping algebras ofsolvable Lie algebras. I will discuss techniques used in joint ongoingwork with Launois (Caen) and Oppong (Greenwich) to understandand describe the derivations and Hochschild cohomology of these andrelated algebras, aiming at discovering new families of analogues ofWeyl algebras and their Poisson counterparts.
Poisson superbialgebras, their representations and
related structures
Abdenacer Makhlouf, abdenacer.makhlouf@uha.fr
The aim of this talk is to introduce the notion of Poisson superbial_x0002_gebra as an analogue of Drinfeld’s Lie superbialgebras. We extendvarious known constructions dealing with representations on Lie su_x0002_perbialgebras to Poisson superbialgebras. We introduce the notionsof Manin triple of Poisson superalgebras and Poisson superbialgebrasand show the equivalence between them in terms of matched pairsof Poisson superalgebras. A combination of the classical Yang-Baxterequation and the associative Yang-Baxter equation is discussed in thisframework. Moreover, we introduce notions of O-operator of weightλ ∈ K of a Poisson superalgebra and post-Poisson superalgebra andinterpret the close relationships between them and Poisson superbial_x0002_gebras. This is a joint work with I. Basdouri, M. Faddous and S.Mabrouk.
Schurification of quantum wreath products
Chun-Ju Lai, cjlai@gate.sinica.edu.tw
Given an algebra A, the Schurification is a procedure that producesan algebra S that enjoys favorable properties similar to the Schuralgebras. I’ll talk about a new theory of Schurification for a vastfamily of algebras admitting a Bernstein-Lusztig presentation. Wedevelop a (geometric) theory of twisted convolution algebras, as wellas an algebraic theory via a Kashiwara-Miwa-Stern tensor space actiontogether with permutation modules. This provides new results forVigneras’ pro-p Iwahori Hecke algebras, the degenerate affine Heckealgebras, and Kleshchev-Muth’s affine zigzag algebras. This is a jointwork with Minets (MPIM).
Some deformed Virasoro algebra and
certaininfinite-dimensional Clifford algebras
Haisheng Li, hli@camden.rutgers.edu
In this talk, we present a natural connection of a deformed Vira_x0002_soro algebra with a certain infinite-dimensional Clifford algebra, byusing the theory of (G, χ)-equivariant ϕ-coordinated quasi modulesfor vertex superalgebras with an automorphism group G and a linearcharacter χ.
Super Duality of Whittaker Modules and Finite
W-algebras
Shun-Jen Cheng, chengsj@math.sinica.edu.tw
We explain an equivalence of categories of Whittaker modules overLie algebras and Lie superalgebras of classical type. These categorieshave properly stratified structures, and under this duality, the simpleand tilting objects correspond. This equivalence is a generalization ofthe so-called super duality between certain parabolic BGG categoriesof Lie algebras and Lie superalgebras of classical type. Using this superduality of Whittaker modules, we establish an equivalence of certaincategories of modules over the corresponding finite W-algebras andW-superalgebras. This is based on a joint work with Weiqiang Wang.
Tensor ideals of abelian type and quantum groups
Kevin Coulembier, kevin.coulembier@sydney.edu.au
The classification of all tensor ideals in (monoidal) representationtheoretic categories is usually an intractable problem. We are inter_x0002_ested in a specific type of tensor ideals that seem to be more tractable.We show how they behave in representation categories of cyclic groupsin positive characteristic and categories of tilting modules over quan_x0002_tum groups at roots of unity. Based on joint work with Pavel Etingofand Victor Ostrik.
The invariant ring of pairs of matrices
Xabier Garc´ıa Mart´ınez, misra@ncsu.edu
Let us consider the action of the general linear group GLn(C) onthe direct product Mdnof d copies of Mn by simultaneous conjuga_x0002_tion sending (X1, ..., Xd) to (gX1g−1, . . . , gXdg−1) for any g ∈ GLn(C).This induces an action of GLn(C) on the algebra C[Mdn] of polyno_x0002_mial functions on Mdn. The algebra of invariants under this action,C[Mdn]GLn, is an important object in several areas of mathematics.In this talk we will explain how we used methods coming fromnon-associative algebras to obtain the full description of the caseC[M24]GL4, which could not be solved using the standard represen_x0002_tation theory methods. Moreover, we will talk about its connectionwith the Calogero-Moser spaces and the Hilbert scheme of points.
The iquantum Brauer category
Alistair Savage, alistair.savage@uottawa.ca
Symmetric pairs consist of a complex simple Lie algebra and a sub_x0002_algebra fixed by an involution. Passing to enveloping algebras, thelatter becomes a Hopf subalgebra. Hence, its category of representa_x0002_tions is naturally a monoidal category. The quantum analogue of thisconcept is that of a quantum symmetric pair. In the quantum setting,the subalgebra, called an iquantum group, is not a Hopf subalgebra.Rather, it is a coideal subalgebra. This means that the category ofrepresentations of the iquantum group is not monoidal. Instead, it isa module category over the category of representations of the largerquantum group. In this talk, we will explore the representation theoryof iquantum groups from the point of view of diagrammatic interpo_x0002_lating categories. We will see that one can obtain a presentation of thecategory of modules of the iquantum group from the framed HOM_x0002_FLYPT skein category (a category that underpins the HOMFLYPTlink invariant) by imposing a single relation. This is joint work withHadi Salmasian and Yaolong Shen.
The solenoidal Virasoro and super-Virasoro
algebras, their constructions and their simple
weight modules
Boujemaa Agrebaoui, b.agreba@fss.rnu.tn
The solenoidal Lie algebra W(n)µ was first introduced by Y. Billigand V. Futorny in [1] where they study its cuspidal modules. In thistalk, we prove that W(n)µ has a one dimensional universal centralextension which is a higher rank Virasoro algebra Vir(n)µ. We willstudy their Vir(n)µ-Harich-Chandra modules. We will give also super_x0002_analogues of W(n)µ and Vir(n)µ and study their weight modules. Inthe end, we use the triangular decomposition of Vir(n)µ given bylexicographic order on Zn. We construct new weight modules havinginfinite dimensional weight spaces.
Two-row Delta Springer varieties
Arik Wilbert, wilbert@southalabama.edu
I will discuss the geometry and topology of a certain family of so_x0002_called Delta Springer varieties from an explicit, diagrammatic point ofview. These singular varieties were introduced by Griffin—Levinson—Wooin 2021 in order to give a geometric realization of an expression thatappears in the t = 0 case of the Delta conjecture of Haglund, Rem_x0002_mel, and Wilson. In the two-row case, Delta Springer varieties gener_x0002_alize both ordinary Springer fibers and Kato’s exotic Springer fibers.Moreover, the homology of two-row Delta Springer varieties has a di_x0002_agrammatic description and can be equipped with an action of thedegenerate affine Hecke algebra. This recovers and upgrades the ac_x0002_tion of the symmetric group obtained by Griffin—Levinson—Woo andyields a skein theoretic description of said action. This is joint workwith A. Lacabanne and P. Vaz.