Mikhail Khovanov, Radmila Sazdanovic
We develop a diagrammatic categorification of the polynomial ring $Z[x]$. Our categorification satisfies a version of Bernstein-Gelfand-Gelfand reciprocity property with the indecomposable projective modules corresponding to $x^n$ and standard modules to $(x-1)^n$ in the Grothendieck ring.
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A categorification of the Jones polynomial
我们构造了一个以Jones多项式为欧拉示性数的连杆的双阶上同调理论。
On a necessary condition for unitary categorification of fusion rings
本文针对 Liu、Palcoux 与 Wu 提出的融合环酉范畴化正不变量必要条件，给出更直接的证明，并讨论这些不变量的整数性质。
Categorification of level two representations of quantum sl(N) via generalized arc rings
我们建立了弧环的扩展，将它们的中心与Springer簇的上同调环联系起来，并对量子sl（N）的所有二级表示进行了分类。
Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial
本文提出了二元Kauffman多项式的SO（2N）单变量特化的一个分类过程。该构造与HOMFLYPT分类有许多相似之处：将多项式的可平面图公式转化为分次向量空间的复形，每个分次向量空间都是与图相关联的Z_2分次微分向量空间的同调，并使用矩阵分解构造。然而，这一次，初等矩阵分解不是Koszul;相反，它们是Koszul矩阵分解的链复形的卷积。我们证明了所得到的复杂的图的链接的同伦类是不变的前两个Reidemeister移动和猜想其不变性下的第三个移动。
Categorifications of the colored Jones polynomial
链接的有色琼斯多项式有两个自然的归一化：一个是n色解结的值为[n+1]，即量子sl（2）的（n+1）维不可约表示的量子维数，另一个是它的值为1。对于每一个正规化，我们构建了一个双阶上同调理论的联系与有色琼斯多项式的欧拉示性数。
Link homology and categorification
这是一个简短的调查代数组合链接同同调论的琼斯多项式和其他链接多项式作为其欧拉示性数。
Let $n$ be a product of two distinct prime numbers. We construct a triangulated monoidal category having a Grothendieck ring isomorphic to the ring of $n$:th cyclotomic integers.
We develop a diagrammatic categorification of the polynomial ring Z[x], based on a geometrically defined graded algebra. This construction generalizes to categorification of some special functions, such as Chebyshev polynomials. Diagrammatic algebras featured in these categorifications lead to the first topological interpretations of the Bernstein-Gelfand-Gelfand reciprocity property.
We introduce colored Jones polynomials of nanowords and their categorification. We also prove the existence of a Khovanov-type bicomplex which has three grades.
Noncommutative Minkowski integral inequality and a unitary categorification criterion for fusion rings
本文证明了非交换闵可夫斯基积分不等式，并给出了融合环的单位ary分类化准则。
A categorification of cyclotomic rings
本研究针对分圆环的范畴化问题，构造了Grothendieck环同构于分圆整数环$\mathbb{O}_n$的三角幺半范畴，为分圆环提供了范畴化实现。
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 …
We use the machinery of categorified Jones-Wenzl projectors to construct a categorification of a type A Reshetikhin-Turaev invariant of oriented framed tangles where each strand is labeled by an arbitrary finite-dimensional representation. As a special case, we obtain a categorification of the coloured Jones polynomial of links.
Using a modified foam evaluation, we give a categorification of the Alexander polynomial of a knot. We also give a purely algebraic version of this knot homology which makes it appear as the infinite page of a spectral sequence starting at the reduced triply graded link homology of Khovanov--Rozansky.
A braid-like isotopy for links in 3-space is an isotopy which uses only those Reidemeister moves which occur in isotopies of braids. We define a refined Jones polynomial and its corresponding Khovanov homology which are, in general, only invariant under braid-like isotopies.
We prove that the 2-variable BMW algebra embeds into an algebra constructed from the HOMFLY-PT polynomial. We also prove that the so(2N)-BMW algebra embeds in the q-Schur algebra of type A. We use these results to construct categorifications of the so(2N)-BMW algebra.
On Khovanov's categorification of the Jones polynomial
职业数学家害怕复杂的文字，但喜欢图片和图表。因此，我们给一个没有花哨的任何图片丰富的一瞥Khovanov的小说建设的“琼斯多项式的分类”。对于相同的低成本，我们还提供了一些计算，包括一个显示，霍瓦诺夫的不变量是严格强于琼斯多项式，并包括一个表的值霍瓦诺夫的不变量的所有主要结多达11个交叉。
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。
We describe a collection of differential graded rings that categorify weight spaces of the positive half of the quantized universal enveloping algebra of the Lie superalgebra gl(m|1).
New Categorifications of the Chromatic and the Dichromatic Polynomials for Graphs
本文对图$G$，定义了多项式环上的分次模链复形\mbox{}，它的分次欧拉示性数与图$G$的色多项式相等.进一步，我们定义了一个双分次模的链复形\mbox{}，它的（双）分次欧拉示性数线等于$G$的重色多项式.这两种构造都使用了Koszul复形，并且类似于HOMFLYPT多项式的新Khovanov-Rozansky范畴\mbox{}。我们还给出了这种三级链节同调论的简单\mbox{}定义的\mbox{}.
There is a $p$-differential on the triply-graded Khovanov--Rozansky homology of knots and links over a field of positive characteristic $p$ that gives rise to an invariant in the homotopy category finite-dimensional $p$-complexes. A differential on triply-graded homology discovered by Cautis is compatible with the $p$-differential structure. As a consequence we get a categorification of the colored Jones polynomial evaluated at a $2p$th root of unity.
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.
This paper classifies the Grothendieck rings of complex fusion categories of multiplicity one up to rank six. Among 72 possible fusion rings, $25$ ones are filtered out by using categorification criteria. Each of the remaining 47 fusion rings admits a unitary complex categorification. We found 6 new Grothendieck rings, categorified by applying a localization approach of the Pentagon Equation.
In this thesis we define and study a categorification of the sl(N)-link polynomial using foams, for N\geq 3. For N=3 we define the universal sl(3)-link homology, using foams, which depends on three parameters and show that it is functorial, up to scalars, with respect to link cobordisms. Our theory is integral. We show that tensoring it with Q yields a theory which is equivalent to the rational universal Khovanov-Rozansky sl(3)-link homology. For N\geq 4 we construct a rational theory categorifying the sl(N)-link polynomial using foams. Our theory is functorial, up to scalars, with respect to …
Categorification of the colored Jones polynomial and Rasmussen invariant of links
我们定义了一个家庭的正式Khovanov括号的彩色链接取决于两个参数。这些括号的同构类是有框有色链接的不变量。Bar-Natan函子应用于这些括号产生Khovanov和李同调论分类有色琼斯多项式。此外，我们研究的条件下，框架彩色链接配边诱导链转换我们的正式括号。我们猜想，对于特殊的参数选择，Khovanov和李同调论的有色链接是函子（最多的符号）。最后，我们将Rasmussen不变量扩展到链接并给出例子，其中该不变量是比多变量Levine-Tristram签名更强的sliceness障碍。
A categorification of finite-dimensional irreducible representations of quantum sl(2) and their tensor products
本文研究sl（2）的量子群的有限维模的张量积的展开。利用复李代数gl（n）的某些Harish-Chandra双模得到了主要的分类.对于单模的特殊情形，我们自然地推出了一个通过某些旗簇的上同调环上的模的范畴化。进一步讨论了它与Steinberg簇的关系和几何代数。我们还给出了一个范畴版本的量化Schur-Weyl对偶和解释的（对偶）典型基地和（对偶）标准基地的投射，倾斜，标准和简单的Harish-Chandra双模。
In this paper, we investigate quantum Fourier analysis on subfactors and unitary fusion categories. We prove the complete positivity of the comultiplication for subfactors and derive a primary $n$-criterion of unitary categorifcation of multifusion rings. It is stronger than the Schur product criterion when $n\geq3$. The primary criterion could be transformed into various criteria which are easier to check in practice even for noncommutative, high-rank, high-multiplicity, multifusion rings. More importantly, the primary criterion could be localized on a sparse set, so that it works for multifu…
We exhibit a monoidal structure on the category of finite sets indexed by P-trees for a finitary polynomial endofunctor P. This structure categorifies the monoid scheme (over Spec N) whose semiring of functions is (a P-version of) the Connes--Kreimer bialgebra H of rooted trees (a Hopf algebra after base change to Z and collapsing H_0). The monoidal structure is itself given by a polynomial functor, represented by three easily described set maps; we show that these maps are the same as those occurring in the polynomial representation of the free monad on P.
There is a known connection between the osp(1|2n) polynomial knot invariant $J_K^n$ and the so(2n+1) knot invariant ${}_{so} J_K^n$ studied by Clark in arXiv:1509.03533 and Blumen in arXiv:0901.3232. In the rank one case, the uncolored $U_{q}(osp(1|2))$ link invariant is equal to the $U_{t^{-1}q}(sl_2)$ link invariant where $t^2=-1$. We define a skein relation similar to the Kauffman bracket, and use that to recover an oriented link invariant which coincides with Clark's uncolored osp(1|2)-link invariant. This definition also comes from the representation theory of $U_{q,π}(sl_2)$, but using d…
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。
An invariant of tangle cobordisms via subquotients of arc rings
我们构造了纠缠对量子sl（2）基本表象的张量幂幂的作用的明确分类。
We construct a bicomplex for the categorification of the colored Jones polynomial. This work is motivated by the problem suggested by Anna Beliakova and Stephan Wehrli who discussed the categorification of the colored Jones polynomial in their paper.
Module Monoidal Categories as Categorification of Associative Algebras
本文引入非单位模幺半范畴的概念以范畴化结合代数的刻画，并证明在有单位情况下该定义与先前模张量范畴定义的2-范畴等价性。
Categorification of the Kauffman bracket skein module of I-bundles over surfaces
Khovanov定义了R ^3中链L的分次同调群，并证明了它们的多项式欧拉示性数是L的Jones多项式。Khovanov的构造并不能直接推广到曲面F而不是D^2上的I-丛M中的链（除了系数仅为Z/2的同调）。因此，本文的目标是提供一个非平凡的推广，他的方法导致同源不变量的链接在M与任意环的系数。在证明了我们的同调群在Reidemeister移动下的不变性之后，我们证明了L的同调群的多项式欧拉示性数决定了L在M的skein模的标准基中的系数.因此，我们的同调群提供了M的Kauffman括号串模的一个“分类”。此外，我们证明了我们的同调群的Viro的正合序列的推广。最后，我们证明了一个对偶定理，它将任意环L的上同调群与L的镜像的同调群联系起来。
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。
We develop a theory of descent and forms of tensor categories over arbitrary fields. We describe the general scheme of classification of such forms using algebraic and homotopical language, and give examples of explicit classification of forms. We also discuss the problem of categorification of weak fusion rings, and for the simplest families of such rings, determine which ones are categorifiable.
On the Colored Jones Polynomial, Sutured Floer homology, and Knot Floer homology
设S^3中的K是一个纽结，\widetilde{K}表示K在其双分支覆盖中的原像，即K的双分支覆盖，K（K）。对任意整数n > 1，证明了从K的镜像的约化n色Jones多项式到（K（K），\widetilde{K}）（当n为奇数）和（S^3，K # K）（当n为偶数）的结弗洛尔同调的Khovanov分类的谱序列的存在性.我们的结果的一个推论是，Khovanov的分类减少n-有色琼斯多项式检测unknot每当n>1。
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。
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.
Categorical lifting of the Jones polynomial: a survey
本文综述了琼斯多项式的范畴化及其在几何、代数和低维拓扑中的意义与影响。
Isomorphisms among quantum Grothendieck rings and cluster algebras
该研究为量子loop代数表示的量子Grothendieck环同构建立cluster理论诠释，得到monoidal范畴化定理的量子化，解决非simply-laced量子loop代数相关长期问题并建立量子$T$-系统，还给出同构对应的显式双有理变换替换公式。
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。
Module categories, weak Hopf algebras and modular invariants
我们发展了幺半范畴上的模范畴的抽象无意义（这是环上模的直接分类）。作为应用，我们证明了任何具有n个单对象的半单幺半范畴等价于弱霍普夫代数的表示范畴（T. Hayashi），并在正整数水平上对$\hat{sl}(2)$的融合范畴上的模块范畴进行分类，在此我们再次遇到ADE分类模式。
In previous work, we have constructed diagrammatic idempotents in an affine extension of the Temperley-Lieb category, which describe extremal weight projectors for sl(2), and which categorify Chebyshev polynomials of the first kind. In this paper, we generalize the construction of extremal weight projectors to the case of gl(N) for N greater than or equal to 2, with a view towards categorifying the corresponding torus skein algebras via Khovanov-Rozansky link homology. As by-products, we obtain compatible diagrammatic presentations of the representation categories of gl(N) and its Cartan subal…
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