A matrix model for quantum SL_2

Ribbon Tableaux, Hall-Littlewood Functions, Quantum Affine Algebras and Unipotent Varieties

We introduce a new family of symmetric functions, which are $q$-analogues of products of Schur functions defined in terms of ribbon tableaux. These functions can be interpreted in terms of the Fock space representation of the quantum affine algebra of type $A_{n-1}^{(1)}$ and are related to Hall-Littlewood functions via the geometry of flag varieties. We present a series of conjectures, and prove them in special cases. The essential step in proving that these functions are actually symmetric consists in the calculation of a basis of highest weight vectors of the $q$-Fock space using ribbon tab…

关于与仿射李代数相关的带状张量范畴的酉性与模性

On the unitarity and modularity of ribbon tensor categories associated with affine Lie algebras

本文通过数值计算和两个猜想，确定了与仿射李代数相关的带状张量范畴在哪些 $q$ 值下为（伪）酉和模的，并与文献结果完全一致，补充了缺失情况。

4维2柄体中ribbon表面的量子不变量

Quantum Invariants of Ribbon Surfaces in $4$-Dimensional $2$-Handlebodies

本文通过单模ribbon范畴构建4维2柄体中ribbon表面的量子不变量，并恢复多个已有不变量，同时推广Reshetikhin-Turaevfunctor到标记Kirby图类，产生带框架链接的不变量。

First fundamental theorems of invariant theory for quantum supergroups

Let $U_q(\mathfrak{g})$ be the quantum supergroup of $\mathfrak{gl}_{m|n}$ or the modified quantum supergroup of $osp_{m|2n}$ over the field of rational functions in $q$, and let $V_q$ be the natural module for $U_q(\mathfrak{g})$. There exists a unique tensor functor, associated with $V_q$, from the category of ribbon graphs to the category of finite dimensional representations of $U_q(\mathfrak{g}$, which preserves ribbon category structures. We show that this functor is full in the cases $\mathfrak{g}=\mathfrak{gl}_{m|n}$ or $osp_{2\ell+1|2n}$. For $\mathfrak{g}=osp_{2\ell|2n}$, we show tha…

Coribbon Hopf (face) algebras generated by lattice models

By studying ``points of the underlying quantum groups''of coquasitriangular Hopf (face) algebras, we construct ribbon categories for each lattice models without spectral parameter of both vertex and face type. Also, we give a classification of the braiding and the ribbon structure on quantized classical groups and modular tensor categories closely related to quantum SU(N)_L-invariants of 3-manifolds.

Ribbon braided module categories, quantum symmetric pairs and Knizhnik-Zamolodchikov equations

Let $\mathfrak u$ be a compact semisimple Lie algebra, and $σ$ be a Lie algebra involution of $\mathfrak u$. Let Rep$_q(\mathfrak u)$ be the ribbon braided tensor C*-category of $U_q(\mathfrak u)$-representations for $0<q<1$. We introduce three module C*-categories over Rep$_q(\mathfrak u)$ starting from the input data $(\mathfrak u,σ)$. The first construction is based on the theory of cyclotomic KZ-equations. The second construction uses the notion of quantum symmetric pair as developed by G. Letzter. The third construction uses a variation of Drinfeld twisting. In all three cases the module …

利用非厄米带环融合范畴构造拓扑量子场论

Constructing TQFTs Using Non-Hermitian Ribbon Fusion Categories

本文提出了一种从非厄米带环融合范畴构造拓扑量子场论的简单方法，并通过证明Müger中心、辫结构和球面结构的相关定理加以验证。

Quantum invariants of framed links from ternary self-distributive cohomology

The ribbon cocycle invariant is defined by means of a partition function using ternary cohomology of self-distributive structures (TSD) and colorings of ribbon diagrams of a framed link, following the same paradigm introduced by Carter, Jelsovsky, Kamada, Langfor and Saito in Transactions of the American Mathematical Society 2003;355(10):3947-89, for the quandle cocycle invariant. In this article we show that the ribbon cocycle invariant is a quantum invariant. We do so by constructing a ribbon category from a TSD set whose twisting and braiding morphisms entail a given TSD $2$-cocycle. Then w…

Ribbon Tableaux and the Heisenberg Algebra

Lascoux, Leclerc and Thibon have introduced symmetric functions which are spin and weight generating functions for ribbon tableaux. This article is aimed at studying these `ribbon functions' in analogy with Schur functions. In particular we will describe ribbon Pieri and Murnagham-Nakayama formulae, a ribbon Cauchy identity and an algebra involution which `conjugates' the ribbon functions. We will study these functions in the context of the action of the Heisenberg algebra on the Fock space representation of the quantum affine algebra U_q(sl_n)^, discovered by Kashiwara, Miwa and Stern. We wil…

Hopf diagrams and quantum invariants

The Reshetikhin-Turaev invariant, Turaev's TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hopf diagrams. This universal encoding is an injective monoidal functor and admits a straightforward monoidal retraction. Any Hopf diagram with n legs yields a n-form on the coend of a ribbon category in a completely explicit way. Thus computing a quantum invariant of a 3-manifold r…

The universal quantum invariant and colored ideal triangulations

The Drinfeld double of a finite dimensional Hopf algebra is a quasi-triangular Hopf algebra with the canonical element as the universal $R$-matrix, and one can obtain a ribbon Hopf algebra by adding the ribbon element. The universal quantum invariant of framed links is constructed using a ribbon Hopf algebra. In that construction, a copy of the universal $R$-matrix is attached to each crossing, and invariance under the Reidemeister III move is shown by the quantum Yang-Baxter equation of the universal $R$-matrix. On the other hand, the Heisenberg double of a finite dimensional Hopf algebra has…

Invariants of Knots and 3-manifolds from Quantum Groupoids

We use categories of representations of finite dimensional quantum groupoids (weak Hopf algebras) to construct ribbon and modular categories that give rise to invariants of knots and 3-manifolds.

Kirby elements and quantum invariants

We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize (in terms of the structure maps of some categorical Hopf algebra) a set of Kirby elements of C which is sufficiently large to recover the known quantum invariants computed from C. The cases of a semisimple ribbon category and of a category of representations are explored in details.

Solving the noncommutative Batalin-Vilkovisky equation

I show that a summation over ribbon graphs with legs gives the construction of the solutions to the noncommutative Batalin-Vilkovisky equation, including the equivariant version. This generalizes the known construction of A-infinity algebra via summation over ribbon trees. These solutions give naturally the supersymmetric matrix action functionals, which are the gl(N)-equivariantly closed differential forms on the matrix spaces, which were introduced in one of my previous papers "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals" (arXiv:0912.5484, electronic CNRS preprint hal-00…

范畴量子对称性与带状张量 2-范畴

Categorical quantum symmetries and ribbon tensor 2-categories

本文证明量子 2-规范变换 Hopf 范畴的 2-表示 2-范畴构成带状张量 2-范畴，并构造平衡函子及 2-缠绕标架不变量。

Quantum double aspects of surface code models

We revisit the Kitaev model for fault tolerant quantum computing on a square lattice with underlying quantum double $D(G)$ symmetry, where $G$ is a finite group. We provide projection operators for its quasiparticles content as irreducible representations of $D(G)$ and combine this with $D(G)$-bimodule properties of open ribbon excitation spaces $L(s_0,s_1)$ to show how open ribbons can be used to teleport information between their endpoints $s_0,s_1$. We give a self-contained account that builds on earlier work but emphasises applications to quantum computing as surface code theory, including…

用Hopf代数构造模张量范畴

Constructing Modular Tensor Categories with Hopf Algebras

本文通过研究Hopf代数的中心扩张、商代数可因子化条件及ribbon元构造方法，得到两类非半单和一族半单的ribbon可因子化Hopf代数，证明其表示范畴为素模张量范畴且无法通过常规张量积构造，为模张量范畴的构建提供了新途径。

Braided homology of quantum groups

We study braided Hochschild and cyclic homology of ribbon algebras in braided monoidal categories, as introduced by Baez and by Akrami and Majid. We compute this invariant for several examples coming from quantum groups and braided groups.

Modified graded Hennings invariants from unrolled quantum groups and modified integral

The second author constructed a topological ribbon Hopf algebra from the unrolled quantum group associated with the super Lie algebra $\mathfrak{sl}(2|1)$. We generalize this fact to the context of unrolled quantum groups and construct the associated topological ribbon Hopf algebras. Then we use such an algebra, the discrete Fourier transforms, a symmetrized graded integral and a modified trace to define a modified graded Hennings invariant. Finally, we use the notion of a modified integral to extend this invariant to empty manifolds and show that it recovers the CGP-invariant.

从不可约精确代数获得带状范畴：简单当前情况

Ribbon categories from ind-exact algebras: simple current case

本文研究了在braided张量范畴的ind-完成中，有限生成局部模是否继承张量范畴结构的条件，并应用于简单当前代数，推导其刚性、braided、ribbon和非退化性质。

四维2-陈-西蒙斯理论的组合量子化II：$D^4$中高阶缎带的量子不变量

Combinatorial quantization of 4d 2-Chern-Simons theory II: Quantum invariants of higher ribbons in $D^4$

本文是四维2-陈-西蒙斯理论组合量子化系列研究的续作，通过刻画加法模*-函子构造非阿贝尔威尔逊面关联，证明其稳定等价性后得到$D^4$中2-缎带的双分次$\u2124$-模量子不变量。

The Twisted Heisenberg Algebra ${\cal U}_{h,w}({\cal H}(4))$

A two parametric deformation of the enveloping Heisenberg algebra ${\cal H}(4)$ which appear as a combination of the standard and a nonstandard quantization given by Ballesteros and Herranz is defined and proved to be Ribbon Hopf algebra. The universal ${\cal R}$-matrix and its associated quantum group are constructed. New solution of Braid group are obtained. The contribution of these parameters in invariants of links and WZW model are analyzed. General results for twisted Ribbon Hopf algebra are derived.

有限维量子群类型超A和非半单模范畴

Finite-dimensional quantum groups of type Super A and non-semisimple modular categories

本文研究了超A类型有限维量子群的构造及其非半单模范畴的分类，通过辫状德林费尔德双和非半单球形结构，提出了新的链接不变量并区分了某些结。

非半单模范畴的模数据

Modular data of non-semisimple modular categories

本文研究非半单模范畴，通过可分解ribbon Hopf代数的表示扩展半单理论，聚焦Cohen-Westreich模数据及其$S$-矩阵对角化混合融合规则，并详细分析小量子群和Nichols Hopf代数实例。

量子映射类群的德尼特旋作用

The Dehn Twist Action for Quantum Representations of Mapping Class Groups

本文研究了量子映射类群中德尼特旋作用的性质，通过分析非分离和分离情况下的顺序，推广了相关结果并探讨了约翰森核与Torelli群的平凡作用。

On a Family of Non-Unitarizable Ribbon Categories

We consider two families of categories. The first is the family of semisimple quotients of H. Andersen's tilting module categories for quantum groups of Lie type $B$ specialized at odd roots of unity. The second consists of categories constructed from a particular family of finite-dimensional quotients of the group algebra of Artin's braid group known as $BMW$-algebras of type $BC$. Our main result is to show that these families coincide as braided tensor categories using a recent theorem of Tuba and Wenzl. The morphism spaces in these categories can be equipped with a Hermitian form, and we a…

Log-modular quantum groups at even roots of unity and the quantum Frobenius I

We construct log-modular quantum groups at even order roots of unity, both as finite-dimensional ribbon quasi-Hopf algebras and as finite ribbon tensor categories, via a de-equivariantization procedure. The existence of such quantum groups had been predicted by certain conformal field theory considerations, but constructions had not appeared until recently. We show that our quantum groups can be identified with those of Creutzig-Gainutdinov-Runkel in type A_1, and Gainutdinov-Lentner-Ohrmann in arbitrary Dynkin type. We discuss conjectural relations with vertex operator algebras at (1,p)-centr…

Non-Semisimple Extended Topological Quantum Field Theories

We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations of unrolled quantum groups, and which can be thought of as a non-semisimple analogue to modular categories. Our approach exploits a 2-categorical version of the universal construction introduced by Blanchet, Habegger, Masbaum, and Vogel. The 1+1+1-EQFTs thus obtained are realize…

Oriented Quantum Algebras and Invariants of Knots and Links

In GT/0006019 oriented quantum algebras were motivated and introduced in a natural categorical setting. Invariants of knots and links can be computed from oriented quantum algebras, and this includes the Reshetikhin-Turaev theory for Ribbon Hopf algebras. Here we continue the study of oriented quantum algebras from a more algebraic perspective, and develop a more detailed theory for them and their associated invariants.

From non-semisimple Hopf algebras to correlation functions for logarithmic CFT

We use factorizable finite tensor categories, and specifically the representation categories of factorizable ribbon Hopf algebras H, as a laboratory for exploring bulk correlation functions in local logarithmic conformal field theories. For any ribbon Hopf algebra automorphism omega of H we present a candidate for the space of bulk fields and endow it with a natural structure of a commutative symmetric Frobenius algebra. We derive an expression for the corresponding bulk partition functions as bilinear combinations of irreducible characters; as a crucial ingredient this involves the Cartan mat…

A quasi-Hopf algebra for the triplet vertex operator algebra

We give a new factorisable ribbon quasi-Hopf algebra U, whose underlying algebra is that of the restricted quantum group for sl(2) at a 2p'th root of unity. The representation category of U is conjecturally ribbon-equivalent to that of the triplet vertex operator algebra W(p). We obtain U via a simple current extension from the unrolled restricted quantum group at the same root of unity. The representation category of the unrolled quantum group is conjecturally equivalent to that of the singlet vertex operator algebra M(p), and our construction is parallel to extending M(p) to W(p). We illustr…

Ribbon tensor structure on the full representation categories of the singlet vertex algebras

We show that the category of finite-length generalized modules for the singlet vertex algebra $\mathcal{M}(p)$, $p\in\mathbb{Z}_{>1}$, is equal to the category $\mathcal{O}_{\mathcal{M}(p)}$ of $C_1$-cofinite $\mathcal{M}(p)$-modules, and that this category admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. Since $\mathcal{O}_{\mathcal{M}(p)}$ includes the uncountably many typical $\mathcal{M}(p)$-modules, which are simple $\mathcal{M}(p)$-module structures on Heisenberg Fock modules, our results substantially extend our previous work on tensor categories of…

An invitation to 2D TQFT and quantization of Hitchin spectral curves

This article consists of two parts. In Part 1, we present a formulation of two-dimensional topological quantum field theories in terms of a functor from a category of Ribbon graphs to the endofuntor category of a monoidal category. The key point is that the category of ribbon graphs produces all Frobenius objects. Necessary backgrounds from Frobenius algebras, topological quantum field theories, and cohomological field theories are reviewed. A result on Frobenius algebra twisted topological recursion is included at the end of Part 1. In Part 2, we explain a geometric theory of quantum curves. …