A diagrammatic approach to categorification of quantum groups III

量子群范畴化的一种图解方法 I

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。

A diagrammatic categorification of the fermion algebra

In this paper, we study the diagrammatic categorification of the fermion algebra. We construct a graphical category corresponding to the one-dimensional fermion algebra, and we investigate the properties of this category. The categorical analogues of the Fock states are some kind of 1-morphisms in our category, and the dimension of the vector space of 2-morphisms is exactly the inner product of the corresponding Fock states. All the results in our categorical framework coincide exactly with those in normal quantum mechanics.

有向图Hecke代数 for Borcherds-Cartan 数据 III: 量子Borcherds超代数的分类化

Quiver Hecke algebras for Borcherds-Cartan datum III: Categorification of quantum Borcherds superalgebras

本文提出了一种有向图Hecke超代数，用于对量子Borcherds超代数进行分类化。

图解代数中的模重写理论及其在范畴化中的应用

Rewriting modulo in diagrammatic algebras and application to categorification

该研究发展了适用于高阶线性结构的模重写理论，将其应用于证明与奇Khovanov同调相关的超2范畴的基猜想，为非专家提供了重写理论导论。

Verma 模的图解范畴化 I：辫子

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。

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。

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…

An introduction to diagrammatic algebra and categorified quantum sl(2)

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.

Diagrammatics for Soergel categories

The monoidal category of Soergel bimodules can be thought of as a categorification of the Hecke algebra of a finite Weyl group. We present this category, when the Weyl group is the symmetric group, in the language of planar diagrams with local generators and local defining relations.

Rewriting modulo isotopies in Khovanov-Lauda-Rouquier's categorification of quantum groups

We study a presentation of Khovanov - Lauda - Rouquier's candidate $2$-categorification of a quantum group using algebraic rewriting methods. We use a computational approach based on rewriting modulo the isotopy axioms of its pivotal structure to compute a family of linear bases for all the vector spaces of $2$-cells in this $2$-category. We show that these bases correspond to Khovanov and Lauda's conjectured generating sets, proving the non-degeneracy of their diagrammatic calculus. This implies that this $2$-category is a categorification of Lusztig's idempotent and integral quantum group $\…

Quantum Frobenius Heisenberg categorification

We associate a diagrammatic monoidal category $\mathcal{H}\textit{eis}_k(A;z,t)$, which we call the quantum Frobenius Heisenberg category, to a symmetric Frobenius superalgebra $A$, a central charge $k \in \mathbb{Z}$, and invertible parameters $z,t$ in some ground ring. When $A$ is trivial, i.e. it equals the ground ring, these categories recover the quantum Heisenberg categories introduced in our previous work, and when the central charge $k$ is zero they yield generalizations of the affine HOMFLY-PT skein category. By exploiting some natural categorical actions of $\mathcal{H}\textit{eis}_k…

幺半范畴化与量子仿射代数 III

Monoidal categorification and quantum affine algebras III

本文对任意量子仿射代数，利用完备对偶数据构造仿射尖点模与行列式模，并证明特定子范畴为量子簇代数提供了幺半范畴化，推广了作者先前的工作。

An equivalence between truncations of categorified quantum groups and Heisenberg categories

We introduce a simple diagrammatic 2-category $\mathscr{A}$ that categorifies the image of the Fock space representation of the Heisenberg algebra and the basic representation of $\mathfrak{sl}_\infty$. We show that $\mathscr{A}$ is equivalent to a truncation of the Khovanov--Lauda categorified quantum group $\mathscr{U}$ of type $A_\infty$, and also to a truncation of Khovanov's Heisenberg 2-category $\mathscr{H}$. This equivalence is a categorification of the principal realization of the basic representation of $\mathfrak{sl}_\infty$. As a result of the categorical equivalences described abo…

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 …