Khovanov弧代数谱化中的谱2-作用、泡沫与框架

A characterisation of indecomposable web-modules over Khovanov-Kuperberg Algebras

After shortly recalling the construction of the Khovanov-Kuperberg algebras, we give a characterisation of indecomposable web-modules. It says that a web-module is indecomposable if and only if one can deduce it directly from the Kuperberg bracket (via a Schur lemma argument). The proofs relies on the construction of idempotents given by explicit foams. These foams are encoded by combinatorial data called red graphs. The key point is to show that when, for a web $w$ the Schur lemma does not apply, one can find an appropriate red graph for $w$.

Sl(N) link homology using foams and the Kapustin-Li formula

We use foams to give a topological construction of a rational link homology categorifying the slN link invariant, for N>3. To evaluate closed foams we use the Kapustin-Li formula adapted to foams by Khovanov and Rozansky. We show that for any link our homology is isomorphic to Khovanov and Rozansky's.

Symmetric Khovanov--Rozansky link homologies

We provide a finite dimensional categorification of the symmetric evaluation of $\mathfrak{sl}_N$-webs using foam technology. As an output we obtain a symmetric link homology theory categorifying the link invariant associated to symmetric powers of the standard representation of $\mathfrak{sl}_N$. In addition, the construction is actually made in an equivariant setting. We prove also that there is a spectral sequence from the Khovanov-Rozansky triply graded link homology to the symmetric one and provide along the way a foam interpretation of Soergel bimodules.

Gl2 Foam Functoriality and Skein Positivity

We prove full functoriality of Khovanov homology for tangled framed gl2 webs. We use this functoriality result to prove a strong positivity result for (orientable) surface skein algebras. The argument goes categorical and consists in proving that so-called linear complexes are stable under superposition.

Singular TQFTs, foams and type D arc algebras

We combinatorially describe the $2$-category of singular cobordisms, called (rank one) foams, which governs the functorial version of Khovanov homology. As an application we topologically realize the type $\mathrm{D}$ arc algebra using this singular cobordism construction.

The diagrammatic Soergel category and sl(N)-foams, for N > 3

For each N > 3, we define a monoidal functor from Elias and Khovanov's diagrammatic version of Soergel's category of bimodules to the category of sl(N) foams defined by Mackaay, Stosic and Vaz. We show that through these functors Soergel's category can be obtained from the sl(N) foams.

The diagrammatic Soergel category and sl(2) and sl(3) foams

We define two functors from Elias and Khovanov's diagrammatic Soergel category, one targeting Clark-Morrison-Walker's category of disoriented sl(2) cobordisms and the other the category of (universal) sl(3) foams.

奇Khovanov同调、高阶表示理论与高阶重写理论

Odd Khovanov homology, higher representation theory and higher rewriting theory

本论文发展了针对奇Khovanov同调的高阶表示理论方法，并提出了适用于高阶代数及其超或分次类似物的重写理论，进而证明了分次$\mathfrak{gl}_2$-泡沫的基定理。

奇霍万诺夫同调与高阶表示理论

Odd Khovanov homology and higher representation theory

本文引入$\boldsymbol{\frak{gl}_2}$-泡沫的超类比物，定义定向缠结的不变量并将其与奇霍万诺夫同调关联，通过构造2阶$q$-舒尔代数的超范畴化及超2-表示，给出奇霍万诺夫同调的表示论构造，还定义了超2-范畴中链复形的张量积。

Khovanov homology and categorification of skein modules

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…

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…

The sl_3 web algebra

In this paper we use Kuperberg's $\mathfrak{sl}_3$-webs and Khovanov's $\mathfrak{sl}_3$-foams to define a new algebra $K^S$, which we call the $\mathfrak{sl}_3$-web algebra. It is the $\mathfrak{sl}_3$ analogue of Khovanov's arc algebra. We prove that $K^S$ is a graded symmetric Frobenius algebra. Furthermore, we categorify an instance of $q$-skew Howe duality, which allows us to prove that $K^S$ is Morita equivalent to a certain cyclotomic KLR-algebra of level 3. This allows us to determine the split Grothendieck group $K^{\oplus}_0(\mathcal{W}^S)_{\mathbb{Q}(q)}$, to show that its center is…

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…