Categorifying fractional Euler characteristics, Jones-Wenzl projector and $3j$-symbols

最小量子余理想与 Jones–Wenzl 投影子的图示法

Diagrammatics for the smallest quantum coideal and Jones--Wenzl projectors

本文研究量子 $\mathfrak{sl}_2$ 张量幂对最小余理想子代数的限制，以弦图和生成元关系刻画其模范畴，并构造 $B/D$ 型 Jones–Wenzl 投影子及 $\Theta$-网络递归公式。

$p$-Jones-Wenzl idempotents

For a prime number $p$ and any natural number $n$ we introduce, by giving an explicit recursive formula, the $p$-Jones-Wenzl projector ${}^p\operatorname{JW}_n$, an element of the Temperley-Lieb algebra $TL_n(2)$ with coefficients in ${\mathbb F}_p$. We prove that these projectors give the indecomposable objects in the $\tilde{A}_1$-Hecke category over ${\mathbb F}_p$, or equivalently, they give the projector in $\mathrm{End}_{\mathrm{SL}_2(\overline{{\mathbb F}_p})}(({\mathbb F}_p^2)^{\otimes n})$ to the top tilting module. The way in which we find these projectors is by categorifying the fra…

带彩色区域的Temperley-Lieb分类和Jones-Wenzl投影子

Temperley-Lieb categories with coloured regions and Jones-Wenzl projectors

本文研究带彩色区域的广义Temperley-Lieb分类，确定其半单性并分解张量积的简单对象，利用双变量Chebyshev多项式和彩色Jones-Wenzl投影子证明了一个猜想。

Symmetric webs, Jones-Wenzl recursions and $q$-Howe duality

We define and study the category of symmetric $\mathfrak{sl}_2$-webs. This category is a combinatorial description of the category of all finite dimensional quantum $\mathfrak{sl}_2$-modules. Explicitly, we show that (the additive closure of) the symmetric $\mathfrak{sl}_2$-spider is (braided monoidally) equivalent to the latter. Our main tool is a quantum version of symmetric Howe duality. As a corollary of our construction, we provide new insight into Jones-Wenzl projectors and the colored Jones polynomials.

The two-color Soergel calculus

We give a diagrammatic presentation for the category of Soergel bimodules for the dihedral group W . The (two-colored) Temperley-Lieb category is embedded inside this category as the degree 0 morphisms between color-alternating objects. The indecomposable Soergel bimodules are the images of Jones-Wenzl projectors. When W is infinite, the parameter q of the Temperley-Lieb algebra may be generic, yielding a quantum version of the geometric Satake equivalence for sl(2). When W is finite, q must be specialized to an appropriate root of unity, and the negligible Jones-Wenzl projector yields the Soe…

Graphical Calculus on Representations of Quantum Lie Algebras

We develop graphical calculation methods. Jones-Wenzl projectors for U_q(sl(2,C)) are very powerful tools to find not only invariants of links but also invariants of 3-manifolds. We find single clasp expansions of generalized Jones-Wenzl projectors for simple Lie algebras of rank 2. Trihedron coefficients of the representation theory for U_q(sl(2,C)) has significant meaning and it is called 3j symbols. Using single clasp expansions for U_q(sl(3,C)), we find some trihedron coefficients of the representation theory of U_q(sl(3,C)). We study representation theory for U_q(sl(4,C)). We conjecture a…

A Lie theoretic categorification of the coloured Jones polynomial

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.

关于着色环状sl(2)纽结同调的统一化

On unification of colored annular sl(2) knot homology

本研究在量子环状Bar-Natan范畴下证明平凡纽结的三种着色sl(2)同调理论等价，通过切比雪夫系统唯一性完成证明，还验证了两类投影算子类的一致性，计算了Khovanov弧代数的量子Hochschild同调，并提出推广至所有纽结的猜想。