Diagrammatic morphisms between indecomposable modules of $\bar{U}_{q}(\mathfrak{sl}_{2})$

Indecomposable 1-morphisms of \dot{U}^+_3 and the canonical basis of U_q^+(sl_3)

We compute the indecomposable objects of \dot{U}^+_3 - the category that categorifies the positive half of the quantum sl_3, and we decompose an arbitrary object into indecomposable ones. On decategorified level we obtain the Lusztig's canonical basis of the positive half U^+_q(sl_3) of the quantum sl_3. We also categorify the higher quantum Serre relations in U_q^+(sl_3), by defining a certain complex in the homotopy category of $\dot{U}^+_3$ that is homotopic to zero. We work with the category $\dot{U}^+_3$ that is defined over the ring of integers. This paper is based on the (extended) diag…

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.

图解非交换 Tangloid 代数的结构与表示

Structure and Representations of a Diagrammatic Non-Commutative Tangloid Algebra

本文引入图解 tangloid 代数及其 braidoid 子代数，通过扩展范畴和约化关系构造双线性配对，为 knotoid 等带端点图解结构提供代数框架。

Categorification of the polynomial ring

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.

定向绞链范畴的典范基

Canonical bases of the oriented skein category

本文通过图示方法为定向绞链范畴的每个态射空间构造杠对合与典范基，并在量子化带墙布劳尔代数上得到Kazhdan-Lusztig型基。

SL(2)倾斜模的箭图

Quivers for SL(2) tilting modules

本文利用图方法定义了一个依赖于素数p的箭图代数，并证明它是特征p下SL(2)倾斜模范畴的底层代数，同时给出了p-Jones-Wenzl投影子之间态射的表示。