A diagrammatic approach to categorification of quantum groups II

量子群范畴化的一种图解方法 I

A diagrammatic approach to categorification of quantum groups I

本文提出一种图解方法，为无环无重边图构造环族，其投射模范畴实现量子群负部分的范畴化。

A diagrammatic categorification of the fermion algebra

In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional fermion algebra, and we investigate the properties of this category. The categorical analogues of the Fock states are some kind of 1-morphisms in our category, and the dimension of the vector space of 2-morphisms is exactly the inner product of the corresponding Fock states. All the results in our categorical framework coincide exactly with those in normal quantum mechanics.

图解代数中的模重写理论及其在范畴化中的应用

Rewriting modulo in diagrammatic algebras and application to categorification

该研究发展了适用于高阶线性结构的模重写理论，将其应用于证明与奇Khovanov同调相关的超2范畴的基猜想，为非专家提供了重写理论导论。

Verma 模的图解范畴化 I：辫子

On Diagrammatic Categorification of Verma Modules I: Braiding

本文通过图解双模的导出张量积范畴化 Verma 模的 R-矩阵，构造辫子函子并证明辫群作用，为纽结补空间建立 Khovanov 同调，并与环形 Khovanov 同调比较。

量子玻色子代数的范畴化

Categorification of quantum boson algebras

本文在任意可对称化Kac-Moody类型下，通过图解生成元与关系构造了量子玻色子代数的分次幺半范畴化，并在Khovanov-Lauda和Rouquier的正部分量子群范畴化上建立了忠实2-表示，进而生成了量子玻色子代数及其扩张的有趣基。

Knot invariants and higher representation theory I: diagrammatic and geometric categorification of tensor products

In this paper, we study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. Our aim is to construct knot homologies categorifying Reshetikhin-Turaev invariants of knots for arbitrary representations, which will be done in a follow-up paper. We consider an algebraic construction of these categories, via an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the quiver Hecke algebra. One of our primary results is that these categories coincide when both are defined. We also inves…

An introduction to diagrammatic algebra and categorified quantum sl(2)

This expository article explains how planar diagrammatics naturally arise in the study of categorified quantum groups with a focus on the categorification of quantum sl2. We derive the definition of categorified quantum sl2 and highlight some of the new structure that arises in categorified quantum groups. The expert will find a discussion of rescalling isomorphisms for categorified quantum sl2, a proof that cyclotomic quotients of the nilHecke algebra are isomorphic to matrix rings over the cohomology ring of Grassmannians, and an interpretation of `fake bubbles' using symmetric functions.

Diagrammatics for Soergel categories

The monoidal category of Soergel bimodules can be thought of as a categorification of the Hecke algebra of a finite Weyl group. We present this category, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations.

Rewriting modulo isotopies in Khovanov-Lauda-Rouquier's categorification of quantum groups

We study a presentation of Khovanov - Lauda - Rouquier's candidate $2$-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to compute a family of linear bases for all the vector spaces of $2$-cells in this $2$-category. We show that these bases correspond to Khovanov and Lauda's conjectured generating sets, proving the non-degeneracy of their diagrammatic calculus. This implies that this $2$-category is a categorification of Lusztig's idempotent and integral quantum group $\…

An odd categorification of quantum sl(2)

We define a 2-category that categorifies the covering Kac-Moody algebra for sl(2) introduced by Clark and Wang. This categorification forms the structure of a super-2-category as formulated by Kang, Kashiwara, and Oh. The super-2-category structure introduces a (Z x Z_2)-grading giving its Grothendieck group the structure of a free module over the group algebra of Z x Z_2. By specializing the Z_2-action to +1 or to -1, the construction specializes to an "odd" categorification of sl(2) and to a supercategorification of osp(1|2), respectively.

Quantum Frobenius Heisenberg categorification

We associate a diagrammatic monoidal category $\mathcal{H}\textit{eis}_k(A;z,t)$, which we call the quantum Frobenius Heisenberg category, to a symmetric Frobenius superalgebra $A$, a central charge $k \in \mathbb{Z}$, and invertible parameters $z,t$ in some ground ring. When $A$ is trivial, i.e. it equals the ground ring, these categories recover the quantum Heisenberg categories introduced in our previous work, and when the central charge $k$ is zero they yield generalizations of the affine HOMFLY-PT skein category. By exploiting some natural categorical actions of $\mathcal{H}\textit{eis}_k…

Virtual braid groups of type B and weak categorification

We define virtual braid groups of type B and construct a morphism from such a group to the group of isomorphism classes of some invertible complexes of bimodules up to homotopy.

An approach to categorification of some small quantum groups II

We categorify an idempotented form of quantum sl2 and some of its simple representations at a prime root of unity.

量子群的内蕴辫子群作用的范畴化 I：2-函子性

Categorification of the internal braid group action for quantum groups I: 2-functoriality

该研究在单李型Kac-Moody代数的范畴化量子群上定义2-函子，实现了Grothendieck群层面Lusztig内蕴辫子群作用的范畴化构造。

Categorification of infinite-dimensional $\mathfrak{sl}_2$-modules and braid group 2-actions I: tensor products

This is the first part of a series of two papers aiming to construct a categorification of the braiding on tensor products of Verma modules, and in particular of the Lawrence--Krammer--Bigelow representations. \\ In this part, we categorify all tensor products of Verma modules and integrable modules for quantum $\mathfrak{sl_2}$. The categorification is given by derived categories of dg versions of KLRW algebras which generalize both the tensor product algebras of Webster, and the dg-algebras used by Lacabanne, the second author and Vaz. We compute a basis for these dgKLRW algebras by using re…

An approach to categorification of some small quantum groups

We categorify one half of the small quantum sl(2) at a prime root of unity. An extension of this construction to an arbitrary simply-laced case is proposed.

量子群可积表示的范畴化

Categorification of integrable representations of quantum groups

本文利用箭图簇上的微局部反常层，对对称量子 Kac–Moody 代数的最高权可积表示及张量积进行范畴化，并几何实现 Lusztig 典范基、得到新的正性结果。

Categorification of integral group rings extended by one dimension

The integral group rings $\mathbb{Z}G$ for finite groups $G$ are precisely those fusion rings whose basis elements have Frobenius-Perron dimension 1, and each is categorifiable in the sense that it arises as the Grothendieck ring of a fusion category. Here we analyze the structure and representation theory of fusion rings with a basis of elements whose Frobenius-Perron dimensions take exactly one value distinct from 1. Our goal is a set of results which assist in characterizing when such fusion rings are categorifiable. As proof of concept, we completely classify the categorifiable near-group …

A Categorification of the Burau representation at prime roots of unity

We construct a p-DG structure on an algebra Koszul dual to a zigzag algebra used by Khovanov and Seidel to construct a categorical braid group action. We show there is a braid group action in this p-DG setting.

Categorification of the adjoint action of quantum groups

Let $U$ be a quantized enveloping algebra. We consider the adjoint action of an $\mathfrak{sl}_2$-subalgebra of $U$ on a subalgebra of $U^+$ that is maximal integrable for this action. We categorify this representation in the context of quiver Hecke algebras. We obtain an action of the 2-category associated with $\mathfrak{sl}_2$ on a category of modules over certain quotients of quiver Hecke algebras. Our approach is similar to that of Kang-Kashiwara for categorifications of highest weight modules via cyclotomic quiver Hecke algebras. One of the main new features is a compatibility of the cat…

拟分裂 i 量子群的分类化

Categorification of quasi-split iquantum groups

本文引入新的分次 2-范畴以推广 2-量子群，并用于对所有对称类型的拟分裂 i 量子群进行分类化。

spin symmetric groups的导出超等价与odd sl(2)-categorifications

Derived superequivalences for spin symmetric groups and odd sl(2)-categorifications

本文利用sl(2)的odd categorification诱导出导出超等价，结合odd 2-category在Hecke-Clifford代数上的作用，证明了spin symmetric group的模表示满足Broue的abelian defect猜想。

A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products

The purpose of this paper is to study categorifications of tensor products of finite dimensional modules for the quantum group for sl(2). The main categorification is obtained using certain Harish-Chandra bimodules for the complex Lie algebra gl(n). For the special case of simple modules we naturally deduce a categorification via modules over the cohomology ring of certain flag varieties. Further geometric categorifications and the relation to Steinberg varieties are discussed. We also give a categorical version of the quantised Schur-Weyl duality and an interpretation of the (dual) canonical …

Junctions of surface operators and categorification of quantum groups

We show how networks of Wilson lines realize quantum groups U_q(sl(m)), for arbitrary m, in 3d SU(N) Chern-Simons theory. Lifting this construction to foams of surface operators in 4d theory we find that rich structure of junctions is encoded in combinatorics of planar diagrams. For a particular choice of surface operators we reproduce known mathematical constructions of categorical representations and categorified quantum groups.

Categorification of Dijkgraaf-Witten Theory

The goal of the paper is to categorify Dijkgraaf-Witten theory, aiming at providing foundation for a direct construction of Dijkgraaf-Witten theory as an Extended Topological Quantum Field Theory. The main tool is cohomology with coefficients in a Picard groupoid, namely the Picard groupoid of hermitian lines.

Cell 2-Representations and Categorification at Prime Roots of Unity

Motivated by recent advances in the categorification of quantum groups at prime roots of unity, we develop a theory of 2-representations for 2-categories enriched with a p-differential which satisfy finiteness conditions analogous to those of finitary or fiat 2-categories. We construct cell 2-representations in this setup, and consider 2-categories stemming from bimodules over a p-dg category in detail. This class is of particular importance in the categorification of quantum groups, which allows us to apply our results to cyclotomic quotients of the categorifications of small quantum group of…

Categorification of the Kauffman bracket skein module of I-bundles over surfaces

Khovanov defined graded homology groups for links L in R^3 and showed that their polynomial Euler characteristic is the Jones polynomial of L. Khovanov's construction does not extend in a straightforward way to links in I-bundles M over surfaces F not D^2 (except for the homology with Z/2 coefficients only). Hence, the goal of this paper is to provide a nontrivial generalization of his method leading to homology invariants of links in M with arbitrary rings of coefficients. After proving the invariance of our homology groups under Reidemeister moves, we show that the polynomial Euler character…

Hall categories and KLR categorification

This paper is the first step in the project of categorifying the bialgebra structure on the half of quantum group $U_{q}(\mathfrak{g})$ by using geometry and Hall algebras. We equip the category of D-modules on the moduli stack of objects of the category $Rep_{\mathbb{C}}(Q)$ of representations of a quiver with the structure of an algebra object in the category of stable $\infty$-categories. The data for this construction is provided by an extension of the Waldhausen construction for the category $Rep_{\mathbb{C}}(Q)$. We discuss the connection to the Khovanov-Lauda-Rouquier categorification o…

Diagrammatic Construction of Representations of Small Quantum $\mathfrak{sl}_2$

We provide a combinatorial description of the monoidal category generated by the fundamental representation of the small quantum group of $\mathfrak{sl}_2$ at a root of unity $q$ of odd order. Our approach is diagrammatic, and it relies on an extension of the Temperley-Lieb category specialized at $δ= -q-q^{-1}$.

Monoidal categorification of cluster algebras II

We prove that the quantum unipotent coordinate algebra $A_q(\mathfrak{n}(w))\ $ associated with a symmetric Kac-Moody algebra and its Weyl group element $w$ has a monoidal categorification as a quantum cluster algebra. As an application of our earlier work, we achieve it by showing the existence of a quantum monoidal seed of $A_q(\mathfrak{n}(w))$ which admits the first-step mutations in all the directions. As a consequence, we solve the conjecture that any cluster monomial is a member of the upper global basis up to a power of $q^{1/2}$.

分类量子群的晶体

The crystal of the categorified quantum group

本文通过无有限性条件的阿贝尔范畴上的范畴作用构造晶体，实现修正量子包络代数晶体的分类化，并识别Webster张量积晶体，进而分类化极值权模。

开Richardson簇上的幺半范畴化

Monoidal categorification on open Richardson varieties

本文证明了在反转冻结簇变量后，quiver Hecke代数模子范畴$\mathscr{C}_{w,v}$构成任意Dynkin型开Richardson簇坐标环的幺半范畴化。

通过加法范畴化解决单项范畴化中的一个问题

Solution of a Problem in Monoidal Categorification by Additive Categorification

本文利用Palu广义变异规则和整体维数至多2的代数相关簇范畴，为Kashiwara-Kim-Oh-Park构造的单项范畴化种子提供了显式箭图，从而解决了量子群表示论与箭图表示论之间的一个簇组合学问题。

Knot invariants and higher representation theory II: the categorification of quantum knot invariants

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor product representations of Kac-Moody algebras and quantum groups, constructed a prequel to this paper. These categories are based on the pictorial approach of Khovanov and Lauda. In this paper, we show that these categories are related by functors corresponding to the braiding and (co…

量子 $ε$ 电Hopf代数与Fock空间范畴化

The quantum $ε$lectrical Hopf algebra and categorification of Fock space

本文引入电KLR代数，证明其范畴化类型A电Lie代数的量子版本，并利用其循环商范畴化更高层Fock空间。

Extended graphical calculus for categorified quantum sl(2)

A categorification of the Beilinson-Lusztig-MacPherson form of the quantum sl(2) was constructed in the paper arXiv:0803.3652 by the second author. Here we enhance the graphical calculus introduced and developed in that paper to include two-morphisms between divided powers one-morphisms and their compositions. We obtain explicit diagrammatical formulas for the decomposition of products of divided powers one-morphisms as direct sums of indecomposable one-morphisms; the latter are in a bijection with the Lusztig canonical basis elements. These formulas have integral coefficients and imply that o…

十二阶模数据的酉范畴化的乘法霍恩障碍

A Multiplicative Horn Obstruction to the Unitary Categorification of a Rank-Twelve Modular Datum

该研究针对低秩分类表中的十二阶潜在模数据，通过分析三股辫群表示等，证明其无法由酉模张量范畴实现，为其酉范畴化提供了乘法霍恩障碍，且论证利用了酉性，非酉范畴化仍开放。

量子仿射代数的表示与特征：在簇范畴化与量子可积模型交叉路口

Representations and characters of quantum affine algebras at the crossroads between cluster categorification and quantum integrable models

本讲座综述无限维量子群表示论与簇范畴化及量子可积模型的联系，并给出新例子和猜想。

Diagrammatics of Braided Group Gauge Theory

We develop a gauge theory or theory of bundles and connections on them at the level of braids and tangles. Extending recent algebraic work, we provide now a fully diagrammatic treatment of principal bundles, a theory of global gauge transformations, associated braided fiber bundles and covariant derivatives on them. We describe the local structure for a concrete $Z_3$-graded or `anyonic' realization of the theory.

Categorification of quantum symmetric pairs I

We categorify a coideal subalgebra of the quantum group of $\mathfrak{sl}_{2r+1}$ by introducing a $2$-category à la Khovanov-Lauda-Rouquier, and show that self-dual indecomposable $1$-morphisms categorify the canonical basis of this algebra. This allows us to define a categorical action of this coideal algebra on the categories of modules over cohomology rings of partial flag varieties and on the BGG category $\mathcal{O}$ of type B/C.

Matrix factorizations and intertwiners of the fundamental representations of quantum group U_q (sl_n)

We want to construct a homological link invariant whose Euler characteristic is MOY polynomial as Khovanov and Rozansky constructed a categorification of HOMFLY polynomial. The present paper gives the first step to construct a categorification of MOY polynomial. For the essential colored planar diagrams with additional data which is a sequence naturally induced by coloring, we define matrix factorizations, and then we define a matrix factorization for planar diagram obtained by gluing the essential colored planar diagrams as tensor product of the matrix factorizations for the essential planar …

An equivalence between truncations of categorified quantum groups and Heisenberg categories

We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $\mathfrak{sl}_\infty$. We show that $\mathscr{A}$ is equivalent to a truncation of the Khovanov--Lauda categorified quantum group $\mathscr{U}$ of type $A_\infty$, and also to a truncation of Khovanov's Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of the principal realization of the basic representation of $\mathfrak{sl}_\infty$. As a result of the categorical equivalences described abo…

Bost-Connes systems, Categorification, Quantum statistical mechanics, and Weil numbers

In this article we develop a broad generalization of the classical Bost-Connes system, where roots of unit are replaced by an algebraic datum consisting of an abelian group and a semi-group of endomorphisms. Examples include roots of unit, Weil restriction, algebraic numbers, Weil numbers, CM fields, germs, completion of Weil numbers, etc. Making use of the Tannakian formalism, we categorify these algebraic data. For example, the categorification of roots of unit is given by a limit of orbit categories of Tate motives while the categorification of Weil numbers is given by Grothendieck's catego…

广义辫群中的范畴化与动力学

Categorification and Dynamics in Generalised Braid Groups

该博士论文受三角范畴与曲面的关联启发，通过范畴化Burau表示将秩2广义辫群实现为三角范畴自等价群，借助范畴熵研究其动力学，用类似映射类群分类的方法完成分类并给出可计算质量增长的算法。

辫子簇的幺范畴化与量子化

Monoidal Categorification and Quantization of Braid Varieties

针对单链型单连通代数群的正辫子字，本文构造辫子簇量子簇代数的幺范畴化，将其与局域化量子环等同，验证特殊化同态恢复原簇代数，还建立了双弦簇变量的量子网格子式与广义量子T-系统。

q-deformations of two-dimensional Yang-Mills theory: Classification, categorification and refinement

We characterise the quantum group gauge symmetries underlying q-deformations of two-dimensional Yang-Mills theory by studying their relationships with the matrix models that appear in Chern-Simons theory and six-dimensional N=2 gauge theories, together with their refinements and supersymmetric extensions. We develop uniqueness results for quantum deformations and refinements of gauge theories in two dimensions, and describe several potential analytic and geometric realisations of them. We reconstruct standard q-deformed Yang-Mills amplitudes via gluing rules in the representation category of t…

幺半范畴化与量子仿射代数 III

Monoidal categorification and quantum affine algebras III

本文对任意量子仿射代数，利用完备对偶数据构造仿射尖点模与行列式模，并证明特定子范畴为量子簇代数提供了幺半范畴化，推广了作者先前的工作。

奇格拉斯曼双模与自旋对称群的导出等价

Odd Grassmannian bimodules and derived equivalences for spin symmetric groups

本文通过奇数格拉斯曼双模理论，证明了自旋对称群的导出等价，并完成Broué的阿贝尔缺陷猜想的证明。