Beijing Institute of Mathematical Sciences and Applications

This is a survey talk including some recent results joint with Xuiao Xu, Hengyi Wang and Rushu Zhuang. One is about the isolcasses of one-parameter exotic quantum small groups at roots of unity. The second is about the descriptions of the Harish-Chandra homomorphism Theorem for two-parameter quantum groups of even rank. If the time permits, I will also talk about the 3rd one, which is about the RLL realizations of two-parameter quantum (affine) algebras of BCD types.

About 40 years ago, Cherednik introduced the parameter-dependent reflection equation. It has seen many applications, especially in quantum integrable systems with boundaries. Its solutions are called K-matrices. So far a universal framework, similar to that for the Yang-Baxter equation and R-matrices, has been lacking. In joint work with A. Appel we describe cylindrical structures on affine quantum groups, building on work by Bao-Wang and Balagovic-Kolb in finite type. These structures extend the quasit

Kitaev's quantum double model has been studied from the lattice model perspective and gauge theory perspective. It can be established from the notions of group algebra and Hopf algebra. Quantum groupoid (a.k.a. weak Hopf algebra) is a generalization of the previous two notions. In this talk, first we will introduce the notion of quantum groupoid and then we will discuss its applications to quantum double models, including the lattice model construction and symmetry breaking based on quantum groupoids.

Given a unitary modular category $\mathcal{C}$, we constructed a $\mathbb{Z}/2\mathbb{Z}$ extension of $\mathcal{C}\boxtimes\mathcal{C}$ which includes all twisted sectors with braidings, generalizing the $\mathbb{Z}/2\mathbb{Z}$ orbifold theory of modular category coming from conformal field theory. It fully generalized the construction of Liu-Xu of a single twisted sector corresponding to $2$-interval Jones-Wassermann subfactor. The applications will also be discussed, this is a joint work with Zhengwei Liu.

Fusion 2-categories are categorification of fusion categories, and they are closely related to the theory of topological phases of matter. For example, (2+1)D topological orders are described by fusion 2-categories while (3+1)D topological orders are described by braided fusion 2-categories. There are natural notions of (commutative) algebras in a (braided) fusion 2-category. The representation theory of these algebras corresponds to string condensations in topological order. Lagrangian algebras are special commutative algebras whose condensed phase is trivial. In this talk, I briefly explain the above notions and give specific examples in $\mathfrak{Z}_1(2Rep(Z_2))$, which corresponds to the (3+1)D toric code. In particular, I give three examples of Lagrangian algebras in this category.

Motivated by the phenomenon that compatible Poisson structures on a cluster algebra play a key role on its quantization (that is, quantum cluster algebra), we introduce the 2nd-stage quantization of a quantum cluster algebra, which means the correspondence between compatible Poisson structures of the quantum cluster algebra and its 2nd-stage quantized cluster algebras. Based on this observation, we find that a quantum cluster algebra possesses a mutually alternating quantum cluster algebra such that their 2nd-stage quantization can be essentially the same. As an example, we give the 2nd-stage quantized cluster algebra $A_{p,q}(SL(2))$ of $Fun_{\C}(SL_{q}(2))$ and show that it is a non-trivial 2nd-stage quantization, which may be realized as a parallel supplement to two parameters quantization of the general quantum group. As another example, we present a class of quantum cluster algebras with coefficients which possess a non-trivial 2nd-stage quantization. In particular we obtain a class of quantum cluster algebras from surfaces with coefficients which possess non-trivial 2nd-stage quantization. Finally, we prove that the compatible Poisson structures of a quantum cluster algebra without coefficients is always a locally standard Poisson structure. Following this, it is shown that the 2nd-stage quantization of a quantum cluster algebra without coefficients is in fact trivial.

Let $R$ be a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$. Let $\mathcal{S}n$ denote the Kauffman bracket skein algebra of the planar surface $\Sigma_{0,n+1}$ over $R$. When $q+q^{-1}$ is invertible in $R$, we find a generating set for $\mathcal{S}n$, and show that the ideal of defining relations is generated by relations of degree at most $6$ supported by certain subsurfaces homeomorphic to $\Sigma_{0,k+1}$ with $k\le 6$. When $q+q^{-1}$ is not invertible, we find another generating set for $\mathcal{S}_n$, and show that the ideal of defining relations is generated by certain relations of degree at most $2n+2$.

We prove that braided fusion categories of Frobenius-Perron $p^mq^nd$ or $p^2q^2r^2$ are weakly group-theoretical, where $p,q,r$ are distinct prime numbers, $d$ is a square-free natural number such that $(pq,d)=1$. As an application, we obtain that weakly integral braided fusion categories of Frobenius-Perron dimension less than $1800$ are weakly group-theoretical, and weakly integral braided fusion categories of odd dimension less than $33075$ are solvable. For the general case, we prove that fusion categories (not necessarily braided) of Frobenius-Perron dimension $84$ and $90$ either solvable or group-theoretical. Together with the results in the literature, this shows that every weakly integral fusion category of Frobenius-Perron dimension less than $120$ is either solvable or group-theoretical. Thus we complete the classification of all these fusion categories in terms of Morita equivalence.

In this talk, I will talk about some progress made in the classification of (pre-)modular fusion categories by global dimension. Explicitly, let $p$ be a prime, by using both of algebraic and arithmetic properties of modular fusion categories, we give a complete classification of non-simple (pre-)modular fusion categories of global dimension $p^2$.

We study the symmetric monoidal 2-category of finite semisimple module categories over a symmetric fusion category. In particular, we study E_n-algebras in this 2-category and compute their $E_n$-centers for $n=0,1,2$. We also compute the factorization homology of stratified surfaces with coefficients given by $E_n$-algebras in this 2-category for $n=0,1,2$ satisfying certain anomaly-free conditions.

In the past thirty years, geometric representation theory has made rapid development, and the geometric approach to quantum algebras is one of directions, which can be regarded as a categorization of quantum algebras. In this talk, I will briefly review some geometric approaches to quantum algebras and recent progress.

In this talk, I will introduce the pseudotensor category and some algebras in a pseudotensor category determined by a Hopf algebra.

This talk deals with the Casimir number for a finite rigid tensor category with finitely many isomorphism classes of indecomposable objects, especially, for a fusion category.

For a finite-dimensional Hopf algebra $A$ with a nonzero left integral $\Lambda$, we investigate a relationship between $P_n(\Lambda)$ and $P_n^J(\Lambda)$, where $P_n$ and $P_n^J$ are respectively the $n$-th Sweedler power maps of $A$ and the twisted Hopf algebra $A^J$. We use this relation to give several invariants of the representation category Rep$(A)$ considered as a tensor category. As applications, we distinguish the representation categories of 12-dimensional pointed nonsemisimple Hopf algebras. Also, these invariants are sufficient to distinguish the representation categories Rep$(K_8)$, Rep$(k Q_8)$ and Rep$(k D_4)$, although they have been completely distinguished by their Frobenius-Schur indicators. We further reveal a relationship between the right integrals $\lambda$ in $A^*$ and $\lambda^J$ in $(A^J)^*$. This can be used to give a uniform proof of the remarkable result which says that the $n$-th indicator $\nu_n(A)$ is a gauge invariant of $A$ for any $n\in \mathbb{Z}$. We also use the expression for $\lambda^J$ to give an alternative proof of the known result that the Killing form of the Hopf algebra $A$ is invariant under twisting. As a result, the dimension of the Killing radical of $A$ is a gauge invariant of $A$.

A well-known open problem is to determine whether an exotic integral quantum symmetry exists, more precisely, whether there is an integral fusion category that is not weakly group-theoretical. In the simple case, we find that this is equivalent to the existence of a non-pointed simple integral modular fusion category. One approach to exploring this question involves searching for simple integral fusion rings to ascertain if one that is not group-like can be categorified. In collaboration with Max Alekseyev, Winfried Bruns, Sebastien Burciu, Linzhe Huang, Zhengwei Liu, Fedor Petrov, Yunxiang Ren, and Jinsong Wu, we have developed several categorification criteria. These involve modular arithmetic, hypergroup theory, quantum Fourier analysis, and a localization strategy related to the pentagon equations. We have used these criteria as effective screening tools for our research. As a result of our studies, we have achieved classification results for the Grothendieck rings of simple integral fusion categories up to rank 8, and for the modular data pertaining to integral modular fusion categories up to rank 12.

The center of an algebra has a universal property identified by Lurie which can be easily generalized to any object in a category with a monoidal structure. Hopf algebras live naturally in a monoidal 2-category by the observation of Street and McCrudden, and we show that the center of a finite dimensional Hopf algebra coincides with the Drinfeld double construction.

Down-up algebras are a special class of hyperbolic ring, cf. A. L. Rosenberg, Noncommutative Algebraic Geometry and Representations of Quantized Algebras, Kluwer, 1995. Differential posets, defined in R. P. Stanley, Differential posets, Journal of the American Mathematical Society (1988), are a special class of partially ordered sets that share many combinatorial and algebraic properties as Young's lattice. The fundamental tool to explore differential posets is a pair of adjoint linear transformations $U$ and $D$, called up and down operator respectively, which have the relation $DU-UD=rI$ for some positive integer $r$. Dual graded graph, developed in S. Fomin, Duality of graded graphs, Journal of Algebraic Combinatorics (1994), is a generalization of differential posets. In this talk, we will discuss some algebraic and combinatorial results related to these concepts.

The notion of Rota-Baxter group of weight 1 was introduced by Guo, Lang and Sheng as the integration of Rota-Baxter Lie algebra of the same weight. How about the classical case of weight 0? On the other hand, Bardakov and Gubarev found that any Rota-Baxter group G of weight 1 associated with its descendent structure naturally produces a skew brace. When is it particularly a brace? A naive solution requiring G to be abelian only gives the trivial brace. In order to answer such two questions, we redefine the notion of Rota-Baxter operator on groups, namely that of weight 0, and justify our new definition especially for Lie groups. Its related algebraic structures, such as relative Rota-Baxter operators on groups, pre-groups and the set-theoretic Yang-Baxter equation, can be discussed correspondingly. This talk is based on joint works with Yunhe Sheng and Rong Tang.

After Witten's interpretation of Jones polynomials in terms of quantum field theory, many topological invariants of low dimensional manifolds were constructed from certain algebraic structures, such as modular tensor categories. In 1990s, Kuperberg and Hennings defined 3-manifold invariants using finite dimensional Hopf algebras. In this talk, we will review their construction and relationship. Then, we will talk about the gauge independence of these topological invariants. That is, they provide algebraic invariants of Hopf algebras.