Hao Zheng
We give a purely geometric categorification of tensor products of finite-dimensional simple $U_q(sl_2)$-modules and $R$-matrices on them. The work is developed in the framework of category of perverse sheaves and the categorification theorems are understood as consequences of Deligne's theory of weights.
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We construct categorifications of tensor products of arbitrary finite-dimensional irreducible representations of $\mathfrak{sl}_k$ with subquotient categories of the BGG category $\mathcal{O}$, generalizing previous work of Sussan and Mazorchuk-Stroppel. Using Lie theoretical methods, we prove in detail that they are tensor product categorifications according to the recent definition of Losev and Webster. As an application we deduce an equivalence of categories between certain versions of category $\mathcal{O}$ and Webster's tensor product categories. Finally we indicate how the categorificati…
We give a geometric categorification of the Verma modules $M(λ)$ for quantum $\mathfrak{sl}_2$.
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 …
In this paper, we prove Khovanov-Lauda's cyclotomic categorification conjecture for all symmetrizable Kac-Moody algebras. Let $U_q(g)$ be the quantum group associated with a symmetrizable Cartan datum and let $V(Λ)$ be the irreducible highest weight $U_q(g)$-module with a dominant integral highest weight $Λ$. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra $R^Λ$ gives a categorification of $V(Λ)$.
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…
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.
Tensor product categorifications, Verma modules and the blob 2-category
本研究构造KLRW代数的dg增强以实现泛sl₂ Verma模与若干可积不可约模张量积的范畴化，进一步在二维可积模情形下构造blob代数的范畴作用并推导其范畴化。
We consider the monoidal subcategory of finite dimensional representations of $U_q(\mathfrak{gl}(1|1))$ generated by the vector representation, and we provide a diagram calculus for the intertwining operators, which allows to compute explicitly the canonical basis. We construct then a categorification of these representations and of the action of both $U_q(\mathfrak{gl}(1|1))$ and the intertwining operators using subquotient categories of the BGG category $\mathcal{O}(\mathfrak{gl}_n)$.
We give a new proof of the "super Kazhdan-Lusztig conjecture" for the Lie super algebra $\mathfrak{gl}_{n|m}(\mathbb{C})$ as formulated originally by the first author. We also prove for the first time that any integral block of category O for $\mathfrak{gl}_{n|m}(\mathbb{C})$ (and also all of its parabolic analogs) possesses a graded version which is Koszul. Our approach depends crucially on an application of the uniqueness of tensor product categorifications established recently by the second two authors.
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。
We introduce a new family of real simple modules over the quantum affine algebras, called the affine determinantial modules, which contains the Kirillov-Reshetikhin (KR)-modules as a special subfamily, and then prove T-systems among them which generalize the T-systems among KR-modules and unipotent quantum minors in the quantum unipotent coordinate algebras simultaneously. We develop new combinatorial tools: admissible chains of i-boxes which produce commuting families of affine determinantial modules, and box moves which describe the T-system in a combinatorial way. Using these results, we pr…
This paper presents categorifications of (right) cell modules and induced cell modules for Hecke algebras of finite Weyl groups. In type $A$ we show that these categorifications depend only on the isomorphism class of the cell module, not on the cell itself. Our main application is multiplicity formulas for parabolically induced modules over a reductive Lie algebra of type $A$, which finally determines the so-called rough structure of generalized Verma modules. On the way we present several categorification results and give the positive answer to Kostant's problem from \cite{Jo} in many cases.…
We introduce and investigate new invariants on the pair of modules $M$ and $N$ over quantum affine algebras $U_q'(\mathfrak{g})$ by analyzing their associated R-matrices. From new invariants, we provide a criterion for a monoidal category of finite-dimensional integrable $U_q'(\mathfrak{g})$-modules to become a monoidal categorification of a cluster algebra.
We suggest a simple definition for categorification of modules over rings and illustrate it by categorifying integral Specht modules over the symmetric group and its Hecke algebra via the action of translation functors on some subcategories of category $\mathcal{O}$ for the Lie algebra $\mathfrak{sl}_n(\mathbb{C})$.
We construct two categorifications of the Lusztig--Vogan module associated to a real reductive algebraic group. The first categorification is given by semisimple complexes in an equivariant derived category, and the second is constructed as a module category over Soergel bimodules. Our categorifications are related by taking equivariant hypercohomology.
Let $U_q(\g)$ be a quantum generalized Kac-Moody algebra and let $V(Λ)$ be the integrable highest weight $U_q(\g)$-module with highest weight $Λ$. We prove that the cyclotomic Khovanov-Lauda-Rouquier algebra $R^Λ$ provides a categorification of $V(Λ)$.
Classification of real modules in monoidal categorifications of cluster algebras
本文提出一个关于实模块最高ℓ-重量单子的猜想公式，并在特定条件下验证了该猜想，同时引入簇模块概念并推广了相关模块的性质。
We give explicit resolutions of all finite dimensional, simple $U_q(\mathfrak{sl_3})$-modules. We use these resolutions to categorify the colored $\mathfrak{sl}_3$-invariant of framed links via a complex of complexes of graded $\mathbb{Z}$-modules.
We categorify Verma and indecomposable projective modules in the category $\mathcal I_{\mathfrak{g}}(\mathfrak{sl}_2)$ for $\mathfrak{sl}_2$ using a tensor product decomposition theorem of T. J. Enright and work of J. Chuang and R. Rouquier, A. Licata and A. Savage (Hecke algebras, finite general linear groups, and Heisenberg categorification) and M. Khovanov (Heisenberg algebra and a graphical calculus).
On uniqueness of tensor products of irreducible categorifications
本文提出张量积范畴化的公理化定义，证明其在标准分层条件下具有唯一性，并给出晶体实现及经典表示论中的例子。
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。
This article investigates duals for bimodule categories over finite tensor categories. We show that finite bimodule categories form a tricategory and discuss the dualities in this tricategory using inner homs. We consider inner-product bimodule categories over pivotal tensor categories with additional structure on the inner homs. Inner-product module categories are related to Frobenius algebras and lead to the notion of $*$-Morita equivalence for pivotal tensor categories. We show that inner-product bimodule categories form a tricategory with two duality operations and an additional pivotal st…
We construct a categorification of parabolic Verma modules for symmetrizable Kac-Moody algebras using KLR-like diagrammatic algebras. We show that our construction arises naturally from a dg-enhancement of the cyclotomic quotients of the KLR-algebras. As a consequence, we are able to recover the usual categorification of integrable modules. We also introduce a notion of dg-2-representation for quantum Kac--Moody algebras, and in particular of parabolic 2-Verma module.
For every oriented surface of finite type, we construct a functorial Khovanov homology for links in a thickening of the surface, which takes values in a categorification of the corresponding gl(2) skein module. The latter is a mild refinement of the Kauffman bracket skein algebra, and its categorification is constructed using a category of gl(2) foams that admits an interesting non-negative grading. We expect that the natural algebra structure on the gl(2) skein module can be categorified by a tensor product that makes the surface link homology functor monoidal. We construct a candidate bifunc…
We explain how to deduce the degenerate analogue of Ariki's categorification theorem over the ground field C as an application of Schur-Weyl duality for higher levels and the Kazhdan-Lusztig conjecture in finite type A. We also discuss some supplementary topics, including Young modules, tensoring with sign, tilting modules and Ringel duality.
We give a definition of monoidal categorifications of quantum cluster algebras and provide a criterion for a monoidal category of finite-dimensional graded $R$-modules to become a monoidal categorification of a quantum cluster algebra, where $R$ is a symmetric Khovanov-Lauda-Rouquier algebra. Roughly speaking, this criterion asserts that a quantum monoidal seed can be mutated successively in all the directions once the first-step mutations are possible. In the course of the study, we also give a proof of a conjecture of Leclerc on the product of upper global basis elements.
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…
We examine the crystal graph of the $\widehat{\mathfrak{sl}}_e$-module arising from an $\widehat{\mathfrak{sl}}_e$-categorification to study the defining endo-functors of the categorification. This yields lower bounds on the number of irreducible constituents of certain objects. We use Ariki's categorification result on Ariki-Koike algebras to obtain a new lower bound on the number of constituents of their parabolically induced modules. In particular this will imply reducibility of every induced module.
We obtain explicit branching rules for graded cell modules and graded simple modules over the endomorphism algebra of a Bott-Samelson bimodule. These rules allow us to categorify a well-known recursive formula for Kazhdan-Lusztig polynomials.
Cluster Categorification of Rank 2 Webs
本文通过网的组合描述，推测性地给出了格拉斯曼流形Gr(k,n)中所有秩2聚类变量的提升。
In recent years different aspects of categorification of the boson-fermion correspondence have been studied. In this paper we propose a categorification of the boson-fermion correspondence based on the category of tensor modules of the Lie algebra $sl(\infty)$ of finitary infinite matrices. By $\mathbb T^+$ we denote the category of "polynomial" tensor $sl(\infty)$-modules. There is a natural "creation" functor $\mathcal T_N: \mathbb T^+\to \mathbb T^+$, $M\mapsto N\otimes M,\quad M,N\in \mathbb T^+$. The key idea of the paper is to employ the entire category $\mathbb T$ of tensor $sl(\infty)$…
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。
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。
We investigate certain categories, associated by Fiebig with the geometric representation of a Coxeter system, via sheaves on Bruhat graphs. We modify Fiebig's definition of translation functors in order to extend it to the singular setting and use it to categorify a parabolic Hecke module. As an application we obtain a combinatorial description of indecomposable projective objects of (truncated) non-critical singular blocks of (a deformed version of) category $\mathcal{O}$, using indecomposable special modules over the structure algebra of the corresponding Bruhat graph.
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。
Categorification of based modules over the complex representation ring of $S_4$
该研究对对称群$S_4$复表示环上秩≤5的不可约有基模分类，得到16个不等价模，并基于此探讨了由子群射影表示诱导的模范畴实现的有基模范畴化问题。
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。
Higher order Kirillov-Reshetikhin modules, Imaginary modules and Monoidal Categorification for $U_q(A_n^{(1)})$
研究量子仿射李代数不可约模族，识别素模并证明唯一分解，推广范畴并定义膨胀，给出虚模存在性及显式公式，并应用于簇变量和 $D_4$ 型虚模。
Real simple modules over simply-laced quantum affine algebras and categorifications of cluster algebras
本文引入简单连通量子仿射代数模范畴的子范畴，证明其量子Grothendieck环具有量子簇代数结构，并据此分类了实单模，提出了相关猜想。
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…
Categorification of generic Su-Zhang character formula
本文提出了一种新的范畴化方法，用于泛泛苏-张特征公式，通过构造基于非共轭博尔子代物的Verma模同态图像的分解，扩展了经典的BGG分解。
The purpose of this paper is to study a categorification of the $n$-th tensor power of the spin representation of $U(\mf{so}(7,\C))$ by using certain singular blocks and projective functors of the BGG category of the complex Lie algebra $\mf{gl}_n$.
A categorification of the Brenti--Welker identity
研究对布伦蒂 - 韦尔克恒等式进行范畴化，核心方法是研究\((\mathbb{C}^r)^{\otimes n}\)张量积分解及相关模，主要贡献是将其从枚举等式提升为对称群表示同构并给出有价值的组合恒等式。
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.
M. Neunh{ö}ffer studies in \cite{Ne} a certain basis of $\mathbb{C}[S_n]$ with the origins in \cite{Lu} and shows that this basis is in fact Wedderburn's basis. In particular, in this basis the right regular representation of $S_n$ decomposes into a direct sum of irreducible representations (i.e. Specht or cell modules). In the present paper we rediscover essentially the same basis with a categorical origin coming from projective-injective modules in certain subcategories of the BGG-category $\mathcal{O}$. An important role in our arguments is played by the dominant projective module in each o…
We introduce a method for associating a chain complex to a module over a combinatorial category, such that if the complex is exact then the module has a rational Hilbert series. We prove homology--vanishing theorems for these complexes for several combinatorial categories including: the category of finite sets and injections, the opposite of the category of finite sets and surjections, and the category of finite dimensional vector spaces over a finite field and injections. Our main applications are to modules over the opposite of the category of finite sets and surjections, known as $FS^{op}$ …
We describe a number of geometric contexts where categorification appears naturally: coherent sheaves, constructible sheaves and sheaves of modules over quantizations. In each case, we discuss how "index formulas" allow us to easily perform categorical calculations, and readily relate classical constructions of geometric representation theory to categorical ones.
Partial $F$-invariants and cluster categorifications
本文研究簇代数中部分$F$-不变量的突变公式，证明Reading猜想，并揭示其与部分$E$-不变量及量子仿射代数极点阶数的联系，给出组合公式。
Parabolic Category O in Complex Rank via Fock Space Tensor Product Categorifications
本文研究一般线性李代数中复秩类比的抛物范畴O，通过Deligne的插值范畴定义。利用sl_Z范畴化理论证明猜想，得到恒定沿面的结构，并通过稳定性极限得到简单对象的特征公式，回答了Etingof的问题。
We give a proof of the periodicity of quantum $T$-systems of type $A_n\times A_\ell$ with certain spiral boundary conditions. Our proof is based on categorification of the $T$-system in terms of the representation theory of quantum affine algebras, more precisely on relations between classes of Kirillov-Reshetikhin modules and of evaluation modules.
Normed modules and the categorification of integrations, series expansions, and differentiations
本文通过为有限维代数上的模赋予范数，构造了两个新范畴来建立积分、级数展开及导数范畴化的框架，并证明了 $\mathscr{A}^p$ 意义下的 Stone--Weierstrass 逼近定理。
We study monoidal categorifications of certain monoidal subcategories $\mathcal{C}_J$ of finite-dimensional modules over quantum affine algebras, whose cluster algebra structures coincide and arise from the category of finite-dimensional modules over quiver Hecke algebra of type A${}_\infty$. In particular, when the quantum affine algebra is of type A or B, the subcategory coincides with the monoidal category $\mathcal{C}_{\mathfrak{g}}^0$ introduced by Hernandez-Leclerc. As a consequence, the modules corresponding to cluster monomials are real simple modules over quantum affine algebras.
Infinite-Dimensional Towers and a Categorification of Differential Operators on $\mathbb{A}^1_{\mathbb{Z}[v,v^{- 1}]}$
本文提出无限维代数塔的公理，通过Grothendieck群的Hopf对偶实现Hesienberg双代数作用并范畴化，最终构造量子微分算子的范畴实现。
The Grothendieck groups of the categories of finitely generated modules and finitely generated projective modules over a tower of algebras can be endowed with (co)algebra structures that, in many cases of interest, give rise to a dual pair of Hopf algebras. Moreover, given a dual pair of Hopf algebras, one can construct an algebra called the Heisenberg double, which is a generalization of the classical Heisenberg algebra. The aim of this paper is to study Heisenberg doubles arising from towers of algebras in this manner. First, we develop the basic representation theory of such Heisenberg doub…
Geometric categorifications of Verma modules: Grassmannian Quiver Hecke algebras
本文通过在 Borel-Moore 同调中引入 Grassmannian-Steinberg quiver flag 簇，将 Naisse-Vaz 扩展 KLR 代数几何实现为卷积代数，从而对 Verma 模进行几何范畴化，并为代数基定理提供了几何解释。
We associate a monoidal category $\mathcal{H}_B$, defined in terms of planar diagrams, to any graded Frobenius superalgebra $B$. This category acts naturally on modules over the wreath product algebras associated to $B$. To $B$ we also associate a (quantum) lattice Heisenberg algebra $\mathfrak{h}_B$. We show that, provided $B$ is not concentrated in degree zero, the Grothendieck group of $\mathcal{H}_B$ is isomorphic, as an algebra, to $\mathfrak{h}_B$. For specific choices of Frobenius algebra $B$, we recover existing results, including those of Khovanov and Cautis--Licata. We also prove tha…
Representations of Brauer category and categorification
该文在特征非2的代数闭域上研究Brauer范畴对应局部代数的表示，用有限Δ-旗模的Grothendieck群范畴化可积最高权𝔰𝔩-模，证明复化后同构，并刻画特征0时标准模与投射覆盖对应的基。
We define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. Our category, which is a q-deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of type A and finite general linear groups. In this way, we obtain a categorification of the bosonic Fock space. We also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting.
In this paper, we give geometric realizations of Lusztig's symmetries. We also give projective resolutions of a kind of standard modules. By using the geometric realizations and the projective resolutions, we obtain the categorification of the formulas of Lusztig's symmetries.
We construct and investigate the structure of the Khovanov-Lauda-Rouquier algebras $R$ and their cyclotomic quotients $R^λ$ which give a categrification of quantum generalized Kac-Moody algebras. Let $U_\A(\g)$ be the integral form of the quantum generalized Kac-Moody algebra associated with a Borcherds-Cartan matrix $A=(a_{ij})_{i,j \in I}$ and let $K_0(R)$ be the Grothedieck group of finitely generated projective graded $R$-modules. We prove that there exists an injective algebra homomorphism $Φ: U_\A^-(\g) \to K_0(R)$ and that $Φ$ is an isomorphism if $a_{ii}\ne 0$ for all $i\in I$. Let $B(…
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