Jiuzu Hong, Antoine Touzé, Oded Yacobi
Fix an infinite field $k$ of characteristic $p$, and let $\g$ be the Kac-Moody algebra $\mathfrak{sl}_{\infty}$ if $p=0$ and $\hat{\mathfrak{sl}}_p$ otherwise. Let $\PP$ denote the category of strict polynomial functors defined over $k$. We describe a $\g$-action on $\PP$ (in the sense of Chuang and Rouquier) categorifying the Fock space representation of $\g$.
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The rings of symmetric polynomials form an inverse system whose limit, the ring of symmetric functions, is the model for the bosonic Fock space representation of the affine Lie algebra. We categorify this construction by considering an inverse limit of categories of polynomial representation of general linear groups. We show that this limit naturally carries an action of the affine Lie algebra (in the sense of Rouquier), thereby obtaining a famiy of categorifications of the bosonic Fock space representation.
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 $\mf{gl}_n$.
We provide a categorical interpretation of a well-known identity from linear algebra as an isomorphism of certain functors between triangulated categories arising from finite dimensional algebras. As a consequence, we deduce that the Serre functor of a finite dimensional triangular algebra A has always a lift, up to shift, to a product of suitably defined reflection functors in the category of perfect complexes over the trivial extension algebra of A.
We categorify various Fock space representations on the algebra of symmetric functions via the category of polynomial functors. In a prequel, we used polynomial functors to categorify the Fock space representations of type A affine Lie algebras. In the current work we continue the study of polynomial functors from the point of view of higher representation theory. Firstly, we categorify the Fock space representation of the Heisenberg algebra on the category of polynomial functors. Secondly, we construct commuting actions of the affine Lie algebra and the level p action of the Heisenberg algebr…
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。
We use geometric parabolic induction functors and the adjoint functors for the supergroups Osp(2m+1,2n) (where m and n vary) to categorify the action of the infinite-dimensional Clifford algebra on the Fock space of semi-infinite forms.
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。
A categorification of the Temperley-Lieb algebra and Schur quotients of U(sl(2)) via projective and Zuckerman functors
我们用基本（二维）sl（2）-模的n次张量幂幂来确定sl（n）的某个最高权范畴的奇异块直和的Grothendieck群. U（sl（2））的作用由投射函子给出，Temperley-Lieb代数的交换作用由Zuckerman函子给出.不可分解的投射函子对应于U（sl（2））中的Lusztig标准基。在对偶实现中，基本表示的n阶张量幂被确定为具有最高权类别的抛物块的直和。跨墙平移函子充当Temperley-Lieb代数的生成元，而Zuckerman函子充当U（sl（2））的生成元。
Whittaker categories, properly stratified categories and Fock space categorification for Lie superalgebras
本文研究I型Lie超代数的Whittaker模范畴，证明其等价于BGG范畴及Harish-Chandra双模奇异范畴的Serre商，并进一步证明这些范畴与Backelin函子一起范畴化了A型量子群的q-对称化Fock空间，其中典范基与倾斜对象、对偶典范基与单对象分别对应。
A diagrammatic presentation of functors and natural transformations and the virtues of biadjointness are discussed. We then review a graphical description of the category of Soergel bimodules and a diagrammatic categorification of positive halves of quantum groups. These notes are a write-up of Takagi lectures given by the author in Hokkaido University in June 2009.
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。
We consider a possible framework to categorify the exponential map exp(-f) given the categorification of a generator f of $\frak{sl}_2$ by Lauda. In this setup the Taylor expansions of exp(-f) and exp(f) turn into complexes built out of categorified divided powers of f. Hom spaces between tensor powers of categorified f are given by diagrammatics combining nilHecke algebra relations with those for a additional "short strand" generator. The proposed framework is only an approximation to categorification of exponentiation, because the functors categorifying exp(f) and exp(-f) are not invertible.
A coherent categorification of the based ring of the lowest two-sided cell
本文给出了仿射 Weyl 群最低双边胞腔基础环 $J_0$ 的部分凝聚层化，并构造复形以提升其基元素及结构常数。
We introduce a non-degenerate bilinear form and use it to provide a new characterization of quantum Kac-Moody superalgebras with no isotropic odd simple roots. We show that the spin quiver Hecke algebras introduced by Kang-Kashiwara-Tsuchioka provide a categorification of half the quantum Kac-Moody superalgebras, using the recent work of Ellis-Khovanov-Lauda. A new idea here is that a super sign is categorified as spin (i.e., the parity-shift functor).
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。
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.
A categorification of representations of $U_q(\mathfrak{gl}_{1|1})$
本文通过构造格拉斯曼流形上向量丛全空间的导出范畴及 Fourier-Mukai 函子，将量子群 $U_q(\mathfrak{gl}_{1|1})$ 在张量积上的作用范畴化。
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 …
We associate a monoidal category $\mathcal{H}^λ$ to each dominant integral weight $λ$ of $\widehat{\mathfrak{sl}}_p$ or $\mathfrak{sl}_\infty$. These categories, defined in terms of planar diagrams, act naturally on categories of modules for the degenerate cyclotomic Hecke algebras associated to $λ$. We show that, in the $\mathfrak{sl}_\infty$ case, the level $d$ Heisenberg algebra embeds into the Grothendieck ring of $\mathcal{H}^λ$, where $d$ is the level of $λ$. The categories $\mathcal{H}^λ$ can be viewed as a graphical calculus describing induction and restriction functors between categor…
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 …
Motivated by the work of of A. Zelevinsky on positive self-adjoint Hopf algebras, we define what we call a symmetric self-adjoint Hopf structure for a certain kind of semisimple abelian categories. It is known that every positive self-adjoint Hopf algebra admits a natural action of the associated Heisenberg double. We construct canonical morphisms lifting the relations that define this action on the algebra level and define an object that we call a categorical Heisenberg double that is a natural setting for considering these morphisms. As examples, we exhibit the symmetric self-adjoint Hopf st…
Moduli Space of Sheaves and Categorified Commutator of Functors
通过稳定层四重模空间的两种交截理论描述，构建了Schiffmann-Vasserot和Neguţ在代数曲面稳定层模空间Grothendieck群上量子环面代数作用的弱范畴化。
We prove that the simple root functors E and F appearing in the categorification of irreducible highest weight modules of quantum groups via cyclotomic Khovanov-Lauda-Rouquier algebras is a biadjoint pair.
We define the i-restriction and i-induction functors on the category O of the cyclotomic rational double affine Hecke algebras. This yields a crystal on the set of isomorphism classes of simple modules, which is isomorphic to the crystal of a Fock space.
Let $p$ be a prime number and let $X$ be a complex algebraic variety with an action of $\mathbb{Z}/p\mathbb{Z}$. We develop the theory of parity complexes in a certain $2$-periodic localization of the equivariant constructible derived category $D^b_{\mathbb{Z}/p\mathbb{Z}}(X,\mathbb{Z}_p)$. Under certain assumptions, we use this to define a functor from the category of parity sheaves on $X$ to the category of parity sheaves on the fixed-point locus $X^{\mathbb{Z}/p\mathbb{Z}}$. This may be thought of as a categorification of Smith theory. When $X$ is the affine Grassmannian associated to some …
Categorification of DAHA and Macdonald polynomials
本文提出与仿射李代数相关的Double Affine Hecke Algebra及其多项式表示和Macdonald多项式的范畴化，通过导出范畴中的函子复合范畴化Demazure算子，并构造模复形使其欧拉示性数等于Macdonald多项式。
The parabolic category $\mathcal{O}$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf{A}$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf{A}$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf{A}$ at level $-N-e$ can be decomposed in terms of the components of the functor $F$ for the…
The algebraic modular functor conjecture in type $A_n$ quantum Teichmüller theory
该研究针对Fock与Goncharov提出的高阶泰希米勒理论量子化的代数模函子猜想，通过拓展Fock-Goncharov框架，证明了$G = \mathrm{PGL}_{n+1}$（即$A_n$型）情形下猜想成立，构造了等变典范切割同构以重构量子代数。
The parabolic category $\mathcal O$ for affine ${\mathfrak{gl}}_N$ at level $-N-e$ admits a structure of a categorical representation of $\widetilde{\mathfrak{sl}}_e$ with respect to some endofunctors $E$ and $F$. This category contains a smaller category $\mathbf A$ that categorifies the higher level Fock space. We prove that the functors $E$ and $F$ in the category $\mathbf A$ are Koszul dual to Zuckerman functors. The key point of the proof is to show that the functor $F$ for the category $\mathbf A$ at level $-N-e$ can be decomposed in terms of components of the functor $F$ for the categor…
Matrix recursion for positive characteristic diagrammatic Soergel bimodules for affine Weyl groups
本文为仿射 Weyl 群构造从图解 Hecke 范畴到矩阵范畴的函子，范畴化递归表示，从而将图解 Soergel 双模约化为按因子 p 更小的双模，并给出 p-典范基及不可分解倾斜模特征标的新下界。
A field-independent filtration of plethystic modules for $\mathrm{SL}_2(\mathbb{F})$ that categorifies a product rule for the Cartan subalgebra of $\mathcal{U}_q(\mathfrak{sl}_2)$
将量子\(\mathfrak{sl}_2\) Cartan子代数的乘积规则提升为\(\mathrm{SL}_2\)仿射群概型的plethystic表示的过滤，这是量子\(\mathfrak{sl}_2\)范畴化的重要一步，通过组合技术利用Weyl模对称函数模型证明。
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