Categorification of some level two representations of sl(n)

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.

A categorification of quantum sl(2)

We categorify Lusztig's version of the quantized enveloping algebra for sl(2). Using a graphical calculus a 2-category is constructed whose split Grothendieck ring is isomorphic to Lusztig's algebra. The indecomposable morphisms of this 2-category lift Lusztig's canonical basis, and the Homs between 1-morphisms are graded lifts of a semilinear form defined on quantum sl(2). Graded lifts of various homomorphisms and antihomomorphisms of Lusztig's algebra arise naturally in the context of our graphical calculus. Using iterated flag varieties, a representation of the 2-category is constructed for…

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。

Categorifications of the colored Jones polynomial

The colored Jones polynomial of links has two natural normalizations: one in which the n-colored unknot evaluates to [n+1], the quantum dimension of the (n+1)-dimensional irreducible representation of quantum sl(2), and the other in which it evaluates to 1. For each normalization we construct a bigraded cohomology theory of links with the colored Jones polynomial as the Euler characteristic.

Tensor products and $q$-characters of HL-modules and monoidal categorifications

We study certain monoidal subcategories (introduced by David Hernandez and Bernard Leclerc) of finite--dimensional representations of a quantum affine algebra of type $A$. We classify the set of prime representations in these subcategories and give necessary and sufficient conditions for a tensor product of two prime representations to be irreducible. In the case of a reducible tensor product we describe the prime decomposition of the simple factors. As a consequence we prove that these subcategories are monoidal categorifications of a cluster algebra of type $A$ with coefficients.

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。

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 via blocks of modular representations for sl(n)

Bernstein, Frenkel, and Khovanov have constructed a categorification of tensor products of the standard representation of $\mathfrak{sl}_2$, where they use singular blocks of category $\mathcal{O}$ for $\mathfrak{sl}_n$ and translation functors. Here we construct a positive characteristic analogue using blocks of representations of $\mathfrak{sl}_n$ over a field $\textbf{k}$ of characteristic $p$ with zero Frobenius character, and singular Harish-Chandra character. We show that the aforementioned categorification admits a Koszul graded lift, which is equivalent to a geometric categorification …

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。

Derived equivalences and sl_2-categorifications for U_q(gl_n)

We give a construction of sl_2-categorifications (in the sense of Chuang-Rouquier) for representations of U_q(gl_n), for generic q and for q a root of unity.

Categorification of Lie algebras [d'apres Rouquier, Khovanov-Lauda]

Given a vector space with an action of a semi-simple Lie algebra, we can try to "categorify" this representation, which means finding a category where the generators of the Lie algebra act by functors. Such categorical representations arise naturally in geometric representation theory and in modular representation theory of symmetric groups. A framework for studying categorical representations was introduced by Rouquier and Khovanov-Lauda. Their definitions are algebraic/diagrammatic, but are connected to the topology of quiver varieties by the work of Rouquier and Varagnolo-Vasserot. In this …

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…

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。

Categorification: tangle invariants and TQFTs

Based on different views on the Jones polynomial we review representation theoretic categorified link and tangle invariants. We unify them in a common combinatorial framework and connect them via the theory of Soergel bimodules. The influence of these categorifications on the development of 2-representation theory and the interaction between topological invariants and 2-categorical structures is discussed. Finally, we indicate how categorified representations of quantum groups on the one hand and monoidal 2-categories of Soergel bimodules on the other hand might lead to new interesting 4-dimen…

A categorification of the skew Howe action on a representation category of $U_q(\mathfrak{gl}(m|n))$

Using quantum skew-Howe duality, we study the category $\operatorname{Rep}(\mathfrak{gl}(m|n))$ of tensor products of exterior powers of the standard representation of $U_q(\mathfrak{gl}(m|n))$, and prove that it is equivalent to a category of ladder diagrams modulo one extra family of relations. We then construct a categorification of $\operatorname{Rep}(\mathfrak{gl}(m|n))$ using the theory of foams. In the case of $n=0$, we show that we can recover $\mathfrak{sl}(m)$ foams introduced by Queffelec and Rose to define Khovanov-Rozansky $\mathfrak{sl}(m)$ link homology. We also define a categor…

Monoidal categorifications of cluster algebras of type A and D

In this note, we introduce monoidal subcategories of the tensor category of finite-dimensional representations of a simply-laced quantum affine algebra, parametrized by arbitrary Dynkin quivers. For linearly oriented quivers of types A and D, we show that these categories provide monoidal categorifications of cluster algebras of the same type. The proof is purely representation-theoretical, in the spirit of [arXiv:0903.1452].

Categorification of the Jones-Wenzl Projectors

The Jones-Wenzl projectors play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes whose graded Euler characteristic is the "classical" projector in the Temperley-Lieb algebra. We show that they are homotopy idempotents and uniquely defined up to homotopy. Our results fit within the general framework of Khovanov's categorification of the Jones polynomial. Consequences of our construction include families of knot invariants corresponding to higher representations of quantum su(2) and…

The Categorified Heisenberg Algebra I: A Combinatorial Representation

We give an introductory account of Khovanov's categorification of the Heisenberg algebra, and construct a combinatorial model for it in a 2-category of spans of groupoids. We also treat a categorification of $U(sl_n)$ in a similar way. These give rise to standard representations on vector spaces through a linearization process.

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…

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。

量子 affine 代数的实模与虚模在 affine 集合代数单子分类中的分类

Classification of real and imaginary modules of quantum affine algebras in monoidal categorifications of affine cluster algebras

本文研究了量子 affine 代数的实模与虚模在 affine 集合代数单子分类中的分类问题，通过组合理论证明了实简单模与簇单项量的对应关系。

A graphical categorification of the two-variable Chebyshev polynomials of the second kind

We show that the $A_2$ clasps in the Karoubi envelope of $A_2$ spider satisfy the recursive formula of the two-variable Chebyshev polynomials of the second kind associated with a root system of type $A_2$. The $A_2$ spider is a diagrammatic description of the representation category for $U_q(\mathfrak{sl}_3)$ and the $A_2$ clasps are projectors. Our categorification also gives a natural definition of a $q$-deformation of the two-variable Chebyshev polynomials. This paper is constructed based only on the linear skein theory and graphical calculus.

Triangular bases in quantum cluster algebras and monoidal categorification conjectures

We consider the quantum cluster algebras which are injective-reachable and introduce a triangular basis in every seed. We prove that, under some initial conditions, there exists a unique common triangular basis with respect to all seeds. This basis is parametrized by tropical points as expected in the Fock-Goncharov conjecture. As an application, we prove the existence of the common triangular bases for the quantum cluster algebras arising from representations of quantum affine algebras and partially for those arising from quantum unipotent subgroups. This result implies monoidal categorificat…

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…

A categorification of U_T sl(1,1) and its tensor product representations

We define the Hopf superalgebra U_T sl(1,1), which is a variant of the quantum supergroup U_q sl(1,1), and its tensor product representations V_1^{\otimes n} for n>0. We construct families of DG algebras A, B and R_n, and consider the DG categories DGP(A), DGP(B) and DGP(R_n), which are full DG subcategories of the categories of DG A-, B- and R_n-modules generated by certain distinguished projective modules. Their 0th homology categories HP(A), HP(B), and HP(R_n) are triangulated and give algebraic formulations of the contact categories of an annulus, a twice punctured disk, and an n times pun…

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…

Categorification and applications in topology and representation theory

This thesis splits into two major parts. The connection between the two parts is the notion of "categorification" which we shortly explain/recall in the introduction. In the first part of this thesis we extend Bar-Natan's cobordism based categorification of the Jones polynomial to virtual links. Our topological complex allows a direct extension of the classical Khovanov complex (h=t=0), the variant of Lee (h=0,t=1) and other classical link homologies. We show that our construction allows, over rings of characteristic 2, extensions with no classical analogon, e.g. Bar-Natan's Z/2Z-link homology…

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 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 …

Categorical geometric skew Howe duality

We categorify the R-matrix isomorphism between tensor products of minuscule representations of U_q(sl(n)) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of U_q(sl(2)) which are related to representations of U_q(sl(n)) by quantum skew Howe duality. The resulting equivalence is part of the program of algebro-geometric categorification of Reshitikhin-Turaev tangle invariants developed by the first two authors.

A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors

We identify the Grothendieck group of certain direct sum of singular blocks of the highest weight category for sl(n) with the n-th tensor power of the fundamental (two-dimensional) sl(2)-module. The action of U(sl(2)) is given by projective functors and the commuting action of the Temperley-Lieb algebra by Zuckerman functors. Indecomposable projective functors correspond to Lusztig canonical basis in U(sl(2)). In the dual realization the n-th tensor power of the fundamental representation is identified with a direct sum of parabolic blocks of the highest weight category. Translation across the…

亏格二DAHA的范畴化

Categorification of the genus two DAHA

该研究在D(2|1,α)电流代数导出范畴上构造三个导出自函子，证明其对应亏格二Macdonald算子，推导算子自伴随性与多项式正交性，完成亏格二DAHA的范畴化。

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 …

Canonical bases and higher representation theory

This paper develops a general theory of canonical bases, and how they arise naturally in the context of categorification. As an application, we show that Lusztig's canonical basis in the whole quantized universal enveloping algebra is given by the classes of the indecomposable 1-morphisms in a categorification when the associated Lie algebra is finite type and simply laced. We also introduce natural categories whose Grothendieck groups correspond to the tensor products of lowest and highest weight integrable representations. This generalizes past work of the author's in the highest weight case…

Notes on link homology

This article consists of six lectures on the categorification of the Burau representation and on link homology groups which categorify the Jones and the HOMFLY-PT polynomial. The notes are based on the lecture course at the PCMI 2006 summer school in Park City, Utah.

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。

Positivity and the canonical basis of tensor products of finite-dimensional irreducible representations of quantum sl(k)

In a categorification of tensor products of fundamental representations of quantum sl(k) via highest weight categories, the indecomposable tilting modules descend to the canonical basis. Since projective functors map tilting modules to tilting modules, the coefficients of the canonical basis of tensor products of finite dimensional, irreducible representations under the action of the Chevalley generators are positive.

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。

Handle Slides and Localizations of Categories

We propose a means by which some categorifications can be evaluated at a root of unity. This is implemented using a suitable localization in the context of prior work by the authors on categorification of the Jones-Wenzl projectors. Within this construction we define objects, invariant under handle slides, which decategorify to the SU(2) quantum invariants at low levels.

An invariant of tangle cobordisms via subquotients of arc rings

We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).

上同调 Hall 代数、其范畴化及其通过挠对的表示

Cohomological Hall algebras, their categorification, and their representations via torsion pairs

本文系统构造上同调、K-理论和范畴化 Hall 代数的表示，统一恢复并范畴化箭图 COHA 及层 COHA 的已知作用，并通过挠对给出新的正负算子作用，同时推广 Khan 的 motivic Borel-Moore 同调以产生新 COHA 例子。

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…

Demazure modules of level two and prime representations of quantum affine $\mathfrak{sl}_{n+1}$

We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting), to $\mathfrak{sl}_{n+1}[t]$--stable, prime Demazure modules in level two integrable highest weight rep…

2-categorical Poincare Representations and State Sum Applications

This is intended as a self-contained introduction to the representation theory developed in order to create a Poincare 2-category state sum model for Quantum Gravity in 4 dimensions. We review the structure of a new representation 2-category appropriate to Lie 2-group symmetries and discuss its application to the problem of finding a state sum model for Quantum Gravity. There is a remarkable richness in its details, reflecting some desirable characteristics of physical 4-dimensionality. We begin with a review of the method of orbits in Geometric Quantization, as an aid to the intuition that th…

Knot invariants and higher representation theory

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 is to study 2-representations of 2-quantum groups (in the sense of Rouquier and Khovanov-Lauda) categorifying tensor products of irreducible representations. These are the representation categories of certain finite dimensional algebras with an explicit diagrammatic presentation, generalizing the cyclotomic quotient of the …

Khovanov homology is a skew Howe 2-representation of categorified quantum sl(m)

We show that Khovanov homology (and its sl(3) variant) can be understood in the context of higher representation theory. Specifically, we show that the combinatorially defined foam constructions of these theories arise as a family of 2-representations of categorified quantum sl(m) via categorical skew Howe duality. Utilizing Cautis-Rozansky categorified clasps we also obtain a unified construction of foam-based categorifications of Jones-Wenzl projectors and their sl(3) analogs purely from the higher representation theory of categorified quantum groups. In the sl(2) case, this work reveals the…

Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras II: Types C_r, F_4, and G_2

We prove the periodicities of the restricted T and Y-systems associated with the quantum affine algebra of type C_r, F_4, and G_2 at any level. We also prove the dilogarithm identities for these Y-systems at any level. Our proof is based on the tropical Y-systems and the categorification of the cluster algebra associated with any skew-symmetric matrix by Plamondon.

定向折叠 Khovanov-Lauda-Rouquier 代数的表示与 Enomoto-Kashiwara 代数

Representations of orientifold Khovanov-Lauda-Rouquier algebras and the Enomoto-Kashiwara algebra

本文提出KLR代数的定向折叠推广，通过Schur-Weyl对偶与量子对称对及Enomoto-Kashiwara代数的最高权模建立联系，给出最高权模的混洗实现与基的构造，并分类标架平凡时的不可约表示及计算整体维数。

Categorifying fractional Euler characteristics, Jones-Wenzl projector and $3j$-symbols

We study the representation theory of the smallest quantum group and its categorification. The first part of the paper contains an easy visualization of the 3j-symbols in terms of weighted signed line arrangements in a fixed triangle and new binomial expressions for the 3j-symbols. All these formulas are realized as graded Euler characteristics. The 3j-symbols appear as new generalizations of Kazhdan-Lusztig polynomials. A crucial result of the paper is that complete intersection rings can be employed to obtain rational Euler characteristics, hence to categorify rational quantum numbers. This …