Advances in Quantum Algebra

February 11 - 14, 2025

Introduction

Quantum algebra has become an important direction in mathematical research in recent years. It is a natural combination and extension of theories such as representation theory, Hopf algebra theory, tensor category theory, and Lie theory. It has also gained widespread attention due to its profound connections with low-dimensional topology, rational conformal field theory, and topological field theory. In recent years, the field of quantum algebra has developed rapidly, with a series of outstanding works and progress emerging.

To enhance communication and discussions among experts in quantum algebra, showcase cutting-edge work from various directions within the field, and promote cross-disciplinary collaborations, we organize the Advances in Quantum Algebra conference in February 2025. The conference focuses on several core directions of quantum algebra, including tensor categories, Hopf algebras, vertex operator algebras, subfactor theory, topological order, topological quantum field theory, and low-dimensional topology, among others.

The conference aims to gather experts’ views and suggestions on core issues in the field, providing valuable references for young scholars in this area. Through the display of cutting-edge work across various directions of quantum algebra, this conference seeks to stimulate intellectual exchange among the experts, providing an excellent platform for cross-disciplinary and cross-field collaboration, while further advancing the development of quantum algebra.

Hosts

Beijing Institute of Mathematical Sciences and Applications (BIMSA)

Tsinghua University

Contact

wyl at bimsa.cn

Venue

Room A-110, TSIMF

Organizers

Zhengwei Liu (刘正伟), BIMSA and Tsinghua University

Sebastien Palcoux, BIMSA

Yilong Wang (王亦龙)，BIMSA

Jinsong Wu (吴劲松), BIMSA

Feb 11, 2025

Opening (8:50 ~ 9:00)
09:00 ~ 09:45	Chongying Dong (董崇英)	Modular extensions of $Rep(G,z)$ and super orbifold theory
09:55 ~ 10:40	Hao Zheng (郑浩)	Modular Extension of Higher Fusion Categories
Tea Break
11:10 ~ 11:55	Li Ren (任丽)	On Kac-Wakimoto hypothesis
Lunch
14:00 ~ 14:45	Gongxiang Liu (刘公祥)	Representation type of Hopf algebras with Chevalley property
14:55 ~ 15:40	Shenglin Zhu (朱胜林)	Yetter-Drinfeld modules over a quasi-triangular Hopf algebras, and its indecomposable module categories decomposition
Tea Break
16:10 ~ 16:55	Liang Chang (常亮)	Modular data of non-semisimple modular categories

Feb 12, 2025

09:00 ~ 09:45	Quanshui Wu (吴泉水)	Numerical Homological Regularities
09:55 ~ 10:40	Hualin Huang (黄华林)	On Symmetric Tensor Decomposition
Tea Break
11:10 ~ 11:55	Zhixiang Wu (吴志祥)	Pseudoalgebras in a homomorphism category
Lunch
14:00 ~ 14:45	Yilong Wang (王亦龙)	Recent progress on (2+1)-alterfold TQFTs
14:55 ~ 15:40	Fan Lu (陆凡)	New Tensor Categories from Sieving Forests
Tea Break and Group Photo

Feb 13, 2025

09:00 ~ 09:45	Naihong Hu (胡乃红)	Quantum Supersymmetries (II): Loewy Filtrations and Quantum de Rham Cohomology
09:55 ~ 10:40	Zhiqiang Yu (于志强)	On the realization of a class of $SL(2,Z)$-representations
Tea Break
11:10 ~ 11:55	Bin Gui (归斌)	From Segal's sewing to pseudo-q-traces and back
Lunch
14:00 ~ 14:45	Li Luo (罗栗)	Geometric construction of quantum Schur algebras
14:55 ~ 15:40	Yunnan Li (黎允楠)	Matched pairs, braiding operators and the Yang-Baxter equation
Tea Break
16:10 ~ 16:55	Haimiao Chen (陈海苗)	A monomial basis for the Kauffman bracket skein algebra of the 4-holed disk
Free Discussions

Feb 14, 2025

09:00 ~ 09:45	Fang Li (李方)	Presentations of mapping class groups and applications to cluster algebras from surfaces
09:55 ~ 10:40	Kun Zhou (周坤)	Quantum Codes Using the τ-OD MP Construction
Tea Break
11:10 ~ 11:55	Hank Chen (陈群皓)	Combinatorial Quantization of 4d derived Chern-Simons theory and a target for ribbon 2-functors
Lunch
14:00 ~ 14:45	Nina Yu (余铌娜)	Fusion Products of Twisted Modules in Pemutation Orbifolds
14:55 ~ 15:40	Kangqiao Li (李康桥)	On the partial dualization of finite-dimensional Hopf algebras: Structures, examples and gauge invariants
End of the Conference

Speaker:: Liang Chang (常亮)
Title:: Modular data of non-semisimple modular categories
Abstract:: Modular tensor categories are semisimple tensor categories with nondegenerated braiding, which have many applications in low dimensional topology and topological physics. Recently, the notion of modularity is extended to non-semisimple tensor category. In this talk, we will talk about the work to extend the well-understood theory of semisimple modular categories, such as the $SL(2,Z)$-representation and rank finiteness, to the non-semisimple case by using representations of factorizable ribbon Hopf algebras.

Speaker:: Haimiao Chen (陈海苗)
Title:: A monomial basis for the Kauffman bracket skein algebra of the 4-holed disk
Abstract:: Fix a commutative ring $R$. Given an oriented surface $F$, its Kauffman bracket skein algebra $\mathcal{S}(F)$ is an $R$-algebra defined in terms of embedded links in $F\times[0,1]$, and is regarded as a quantization of the $SL(2,\mathbb{C})$-character variety of $\pi_1(F)$. It is well-known that the multi-curves in $F$ form a basis for $\mathcal{S}(F)$ as a free $R$-module. For $F$ being the 4-holed disk, we present a monomial basis for $\mathcal{S}(F)$, which is more convenien in algebraic manipulations. We apply the result to compute the Kauffman bracket skein module of a knot exterior, an show a nontrivial phenomenon.

Speaker:

Hank Chen (陈群皓)

Title:

Combinatorial Quantization of 4d derived Chern-Simons theory and a target for ribbon 2-functors

Abstract:

The procedures for producing 3d TQFTs from the data of quantum group Hopf algebras are well-known. In order to lift these constructions to higher-dimensions, it is widely accepted that we need to perform a categorification. However, many unknowns and obstacles still stand in our way. As a first step towards a resolution, I will introduce a derived higher-homotopy generalization of the 3d Chern-Simons theory to 4-dimensions, and describe the higher-categorical framework in which the we can quantize it on a lattice. Based on the geometry of stratified 3-manifolds and its 2-skeleton, I will show how we can extract data from the underlying 4d action in order to equip the discrete degrees-of-freedom of 2-Chern-Simons holonomies with the structure of a Hopf cocategory.

Then, if time permits, I will discuss the representation 2-category of its quantum gauge symmetries, and demonstrate that it is ribbon tensor (and what this means). This servers as the target for a ribbon 2-functor from the Baez-Langford 2-category of 2-tangles, which completes by the 2-tangle hypothesis into a 4d “2-Chern-Simons TQFT”.

This is based on my recent works https://arxiv.org/abs/2501.06486 and https://arxiv.org/abs/2501.08041.

Speaker:: Chongying Dong (董崇英)
Title:: Modular extensions of $Rep(G,z)$ and super orbifold theory
Abstract:: For any finite group G, its module category Rep(G) is a symmetric braided fusion category. The modular extensions of Rep(G) have beenstudied extensively by Muger, Drinfeld-Gelaki-Nikshych-Ostrik, Lan-Kong-Wen and others. If z is an order 2 central element, G-module category Rep(G,z) with a new braiding determined by z is also a symmetric braided fusion category, but the modular extensions of Rep(G,z) have been only known for several examples. This talk will report our recent progress in understanding the modular extensions of Rep(G,z) by using the super orbifold theory. This is a joint work with Richard Ng, Li Ren and Yilong Wang.

Speaker:: Bin Gui (归斌)
Title:: From Segal’s sewing to pseudo-q-traces and back
Abstract:: In 1990, Zhu proved that if V is a C2 cofinite rational VOA, then the q-traces of the vertex operators for modules of V span a modular-invariant space. These q-traces have a clear geometric meaning: they are special cases of Segal’s sewing construction (≈partial contractions for conformal blocks). However, if V is C2 cofinite but irrational, Miyamoto proved in 2004 that achieving modular invariance requires generalizing q-traces to pseudo-q-traces. At first glance, pseudo-q-traces do not appear to fit within Segal’s sewing framework. Did Segal miss something? In this talk I will provide the answer: No. By suitably adjusting Segal’s sewing, we can achieve a geometric interpretation of pseudo-q-traces. This is joint work with Hao Zhang.

Speaker:: Naihong Hu (胡乃红)
Title:: Quantum Supersymmetries (II): Loewy Filtrations and Quantum de Rham Cohomology
Abstract:: We explore the indecomposable submodule structure of quantum Grassmann super-algebra $\Omega_q(m|n)$ and its truncated objects $\Omega_q(m|n,\textbf{r})$ in the case when $q=\varepsilon$ is an $\ell$-th root of unity. A net-like weave-lifting method is developed to show the indecomposability of all the homogeneous super subspaces $\Omega_q^{(s)}(m|n,\textbf{r})$ and $\Omega_q^{(s)}(m|n)$ as $\mathcal U_q(\mathfrak{gl}(m|n))$-modules by defining “energy grade” to depict their “$\ell$-adic” phenomenon. Their Loewy filtrations are described, the Loewy layers and dimensions are determined by combinatorial identities. The quantum super de Rham cochain short complex $(\mathcal D_q(m|n)^{(\bullet)},d^\bullet)$ is constructed and proved to be acyclic (Poincar'e Lemma), where $\mathcal D_q(m|n)=\Omega_q(m|n)\otimes \sqcap_q(m|n)$ and $\sqcap_q(m|n)$ is the quantum exterior super-algebra, over which we define the $q$-differentials such that the product structure of $\sqcap_q(m|n)$, the quantum exterior super-algebra, is well-matched everywhere. However, the truncated quantum de Rham cochain subcomplexes $(\mathcal D_q(m|n,\textbf{r})^{(\bullet)},d^\bullet)$ we mainly consider are no longer acyclic and calculating the resulting quantum super de Rham cohomologies $H^s_{DR}(\mathcal D_q(m|n, \mathbf r)^{(\bullet)})$ are highly nontrivial. This is a joint work with Dr. Ge Feng and Prof. Marc Rosso.

Speaker:: Hualin Huang (黄华林)
Title:: On Symmetric Tensor Decomposition
Abstract:: In this talk, we give a brief introduction to symmetric tensor decomposition. One of the main approaches involves a natural Hopf pairing between polynomial rings and rings of differential operators. The idea is elucidated by explicit examples of binary forms.

Speaker:: Yunnan Li (黎允楠)
Title:: Matched pairs, braiding operators and the Yang-Baxter equation
Abstract:: 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, namely matched pair and braiding operator, etc, constructed by groups and also Hopf algebras, for finding set-theoretic type solutions to the Yang-Baxter equation, and then discuss some related problems being concerned.

Speaker:

Fang Li (李方)

Title:

Presentations of mapping class groups and applications to cluster algebras from surfaces

Abstract:

In this talk, we give presentations of the mapping class groups of marked surfaces stabilizing boundaries for any genus. Note that in the existing works, the mapping class groups of marked surfaces were the isotopy classes of homeomorphisms fixing boundaries pointwise. The condition for stabilizing boundaries of mapping class groups makes the requirement for mapping class groups to fix boundaries pointwise to be unnecessary.

As an application of presentations of the mapping class groups of marked surfaces stabilizing boundaries, we obtain the presentation of the cluster automorphism group of a cluster algebra from a feasible surface (S,M).

Lastly, for the case (1) 4-punctured sphere, the cluster automorphism group of a cluster algebra from the surface is characterized. Since cluster automorphism groups of cluster algebras from those surfaces were given in the cases (2) the once-punctured 4-gon and (3) the twice-punctured digon, we indeed give presentations of cluster automorphism groups of cluster algebras from surfaces which are not feasible.

This is a joint work with Jinlei Dong.

Speaker:

Kangqiao Li (李康桥)

Title:

On the partial dualization of finite-dimensional Hopf algebras: Structures, examples and gauge invariants

Abstract:

Let H be a finite-dimensional Hopf algebra with left coideal subalgebraB. There is a structure of quasi-Hopf algebra on the smash product $(H / B^{+} H )^* \# B$, which is called the left partially dualized quasi-Hopf algebra (or partial dual for short) of H for B. It reconstructs the dual tensor category of Rep(H) with respect to its left module category Rep(B), where Rep(H) and Rep(B) denote categories of finite-dimensional representations of respective algebras.

In this talk, we introduce the construction of partial duals, as well as:

Conditions when partial duals are particular;
Formulations of partial duals for some classical structures, including bismash products, bosonizations and quantum doubles;
Descriptions for gauge invariants of partial duals, such as indicators and exponent.

Speaker:: Gongxiang Liu (刘公祥)
Title:: Representation type of Hopf algebras with Chevalley property
Abstract:: In this talk, we try to talk about Hopf algebras with Chevalley property of finite, tame and discrete corepresentation typies. This is a joint work with Yu Jing.

Speaker:: Fan Lu (陆凡)
Title:: New Tensor Categories from Sieving Forests
Abstract:: We discovered several new examples of tensor categories from our classification results on fusion bialgebras with exchange relations, one of which is the binary tree tensor category. This classification scheme begins by transforming infinite diagrammatic consistency equations of exchange relations into a finite set of algebraic equations of degree at most 3. We then introduce a key concept, the fusion graph of a fusion bialgebra, and prove that the fusion graph for any minimal projection is a forest if and only if the fusion bialgebra has an exchange relation. For each forest fusion graph, the system of degree 3 equations reduces to linear and quadratic equations that are efficiently solvable. To deal with exponentially many forest fusion graphs in the unitary case, we propose two analytic criteria to sieve most candidates from subfactor planar algebras. We developed a computer program to compute the Grobner basis of the consistency equations for each forest fusion graph up to 6-dimension, and to sieve forest fusion graphs using our criteria without solving the equations. New examples of fusion bialgebras with exchange relations corresponding to new tensor categories were found in this process. The sieving criteria are much more efficient than directly solving the equations, demonstrating the advantages of quantum Fourier analysis.

Speaker:: Li Luo (罗栗)
Title:: Geometric construction of quantum Schur algebras
Abstract:: We provide the geometric construction of a series of generalized Schur algebras of any type via Borel-Moore homologies and equivariant K-groups of generalized Steinberg varieties. As applications, we obtain a Schur algebra analogue of the local geometric Langlands correspondence of any type, provide an equivariant K-theoretic realization of quasi-split i-quantum groups of affine type AIII, and establish a geometric Howe duality for affine i-quantum groups.

Speaker:: Li Ren (任丽)
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.

Speaker:: Yilong Wang (王亦龙)
Title:: Recent progress on (2+1)-alterfold TQFTs
Abstract:: In this talk, we will give an overview of our joint program with Zhengwei Liu, Shuang Ming and Jinsong Wu on the development of the (2+1)-alterfold theory. We will demonstrate how concepts in the familiar categorical constructions of (2+1)-TQFTs are topologized and unified in the framework of alterfolds, which in particular results in a quick proof of the “RT=TV” type of result. We will then show how Morita contexts can be naturally included in the alterfold theory. Finally, if permits, we will explain how major results on the alpha-induction construction follows from intuititve alterfold calculus and how we generalize them based on such intuition.

Speaker:: Quanshui Wu (吴泉水)
Title:: Numerical Homological Regularities
Abstract:: Inspired by the studies in algebraic geometry and commutative algebra, Jorgensen first defined CM-regularities for graded modules over noncommutative noetherian connected graded algebras. Two fundamental commutative results are generalized to the non-commutative case: a vanishing-theorem by Mumford, and a theorem on linear resolutions and syzygies by Eisenbud and Goto. Subsequently, Jorgensen established two inequalities relating CM-regularity to Tor-regularity, which spurred numerous intriguing research efforts. For example, Romer gave a characterization that a commutative standard graded algebra is a polynomial algebra if and only if either of the two of Jorgensen’s inequalities is always an equality for any finitely generated graded module. Dong and Wu generalized Romer’s result, and showed that the CM-regularity of an algebra A can be considered as an invariant that measures how far away A is from being AS-regular for any Koszul noetherian connected graded algebra A with a balanced dualizing complex. In the last two years, Kirkman-Won-Zhang did a lot of work about the regularities, in particular they gave a far-reaching generalization of Dong-Wu’s result by introducing another numerical homological invariant ASreg(A) for any noetherian connected graded k-algebra A. Recently, Wu and Yi defined more numerical homological invariants over positively graded algebras and studied relations between them. I will concentrate to Wu-Yi’s work in the talk.

Speaker:: Zhixiang Wu (吴志祥)
Title:: Pseudoalgebras in a homomorphism category
Abstract:: We mainly introduce an associative algebra in a new pseudotensor category based on a homomorphism category of left $H$-modules, where $H$ is a Hopf algebra. This algebra is called an associative $\mathcal{H}H$-pseudoalgebra, which consists of an associative $H$-pseudoalgebra and a bipseudomodule over this associative $H$-pseudoalgebra. Basic properties, extensions of representations of an associative $\mathcal{H}H$-pseudoalgebra are described by a group and cohomology of associative $\mathcal{H}H$-pseudoalgebras is established.

Speaker:: Zhiqiang Yu (于志强)
Title:: On the realization of a class of $SL(2,Z)$-representations
Abstract:: In this talk, I will talk about the realization of a class of irreducible representation of the modular group $SL(2,Z)$. Explicitly, let $p<q$ be odd primes, $\rho_1$ and $\rho_2$ be irreducible representations of $SL(2,Z/p)$ and $SL(2,Z/q)$ of dimensions $\frac{p+1}{2}$ and $\frac{q+1}{2}$, respectively. If $\rho_1\oplus\rho_2$ can be realized as modular representation associated to a modular fusion category $\mathcal{C}$, then $q-p=4$. Moreover, if $\mathcal{C}$ contains a non-trivial '{e}tale algebra, then $\mathcal{C}$ is connected with a near-group fusion category of type $Z/p+p$, which gives a partial answer to the conjecture of D. Evans and T. Gannon. By using the Witt equivalence of fusion categories, I also introduce an infinite classes of potential realizable fusion rings which contain elements of Frobenius-Perron dimension $\frac{\sqrt{p}+\sqrt{q}}{2}$.

Speaker:: Nina Yu (余铌娜)
Title:: Fusion Products of Twisted Modules in Pemutation Orbifolds
Abstract:: Orbifold theory explores the structure of vertex operator algebras (VOAs) when finite groups act on them, with a particular focus on understanding the representation theory of the fixed point subalgebra. A key aspect of this field is permutation orbifolds, which study how the symmetric group acts on the n-fold tensor product of a VOA. In this talk, I will present our recent findings on the fusion product of twisted modules in permutation orbifolds. This work is a collaboration with C. Dong and F. Xu.

Speaker:: Hao Zheng (郑浩)
Title:: Modular Extension of Higher Fusion Categories
Abstract:: The concept of modular extension, first introduced by Muger, has proven to be intimately related to the symmetries of topological phases. In this talk, we provide a higher formulation of modular extension and present several classification results. Notably, we demonstrate the existence of a surprising long exact sequence formed by groups of modular extensions and quantum Witt groups.

Speaker:: Kun Zhou (周坤)
Title:: Quantum Codes Using the τ-OD MP Construction
Abstract:: We propose a method called the τ-optimal defining (τ-OD) matrix-product (MP) construction to derive infinite families of quantum codes with good parameters. Through this scheme, we present 100 record-breaking quantum codes, which exceed the best-known lower bounds on the minimum distances of quantum codes listed in Grassl’s online database (Joint with Meng Cao).

Speaker:: Shenglin Zhu (朱胜林)
Title:: Yetter-Drinfeld modules over a quasi-triangular Hopf algebras, and its indecomposable module categories decomposition
Abstract:: Let G be a finite group and $C=_{G}^{G}YD$ be the corresponding tensor category of Yetter-Drinfeld modules. Then the conjugacy class decomposition of G interprets C as a direct sum of indecomposable C-module categories, with each such indecomposable C-module category itself being a tensor category, which is isomorphic to a module category over a subgroup algebra of G. In this talk we extended this result to finite dimensional quasi-triangular Hopf algebras, and try to find mechanism to reconstruct the tensor category $_{H}^{H}YD$ from the indecomposable module categories.