Categories arising from tabular algebras

Localization of $\frak{u}$-modules. III. Tensor categories arising from configuration spaces

This article is a sequel to hep-th/9411050, q-alg/9412017. In Chapter 1 we associate with every Cartan matrix of finite type and a non-zero complex number $ζ$ an abelian artinian category $\FS$. We call its objects {\em finite factorizable sheaves}. They are certain infinite collections of perverse sheaves on configuration spaces, subject to a compatibility ("factorization") and finiteness conditions. In Chapter 2 the tensor structure on $\FS$ is defined using functors of nearby cycles. It makes $\FS$ a braided tensor category. In Chapter 3 we define, using vanishing cycles functors, an exact …

Derived Categories and Lie Algebras

Let $\md^b(A)$ be the derived category of a finite dimensional basic algebra $A$ with finite global dimension. We construct the Lie algebra arising from the 2-periodic version $\mk_2(\mp(A))$ of $\mk^b(\mp(A))$ in term of constructible functions on varieties attached to $\mk_2(\mp(A))$.

Meromorphic tensor categories

We introduce the notion of meromorphic tensor category and illustrate it in several examples. They include representations of quantum affine algebras, chiral algebras of Beilinson and Drinfeld, G-vertex algebras of Borcherds, and representations of GL over a local field. Hopefully the formalism will accomodate various tensor structures arising in relation to the quantized Knizhnik-Zamolodchikov equations and deformed CFT

紧致半单张量2-范畴是Morita连通的

Compact Semisimple Tensor 2-Categories are Morita Connected

本文证明紧致半单张量2-范畴在任意特征零域上具有Morita连通性，推广了融合2-范畴的相应结果，并由此构造了由第四Galois上同调群索引的辫融合1-范畴。

Prime decomposition of modular tensor categories of local modules of Type D

Let $\mathcal{C}(\mathfrak{g},k)$ be the unitary modular tensor categories arising from the representation theory of quantum groups at roots of unity for arbitrary simple finite-dimensional complex Lie algebra $\mathfrak{g}$ and positive integer levels $k$. Here we classify nondegenerate fusion subcategories of the modular tensor categories of local modules $\mathcal{C}(\mathfrak{g},k)_R^0$ where $R$ is the regular algebra of Tannakian $\text{Rep}(H)\subset\mathcal{C}(\mathfrak{g},k)_\text{pt}$. For $\mathfrak{g}=\mathfrak{so}_5$ we describe the decomposition of $\mathcal{C}(\mathfrak{g},k)_R^…

源自3C代数的单位元顶点算子代数

A unitary vertex operator algebra arising from the 3C-algebra

本文通过3C代数的共轭表示证明了特定顶点算子代数的单位性，并推导了其模的融合规则。

划分范畴中的算法问题

Algorithmic Problems in Categories of Partitions

本文研究划分范畴的计算问题，提出高效算法与数据结构并在OSCAR中实现，同时证明存在划分范畴使成员判定与规模计数问题在算法上不可判定。

关于由Levin-Wen模型产生的弱Hopf代数的表示范畴

On the Representation Categories of Weak Hopf Algebras Arising from Levin-Wen Models

本文在去除酉性条件下独立证明了Kitaev-Kong弱Hopf代数的表示范畴与相应模函子范畴幺半等价，并由此构造出表示范畴辫子等价于Drinfeld中心的拟三角弱Hopf代数，同时澄清了其与管代数的Morita等价关系在一般情形下无法提升为幺半等价。

Internal algebra classifiers as codescent objects of crossed internal categories

Inspired by recent work of Batanin and Berger on the homotopy theory of operads, a general monad-theoretic context for speaking about structures within structures is presented, and the problem of constructing the universal ambient structure containing the prescribed internal structure is studied. Following the work of Lack, these universal objects must be constructed from simplicial objects arising from our monad-theoretic framework, as certain 2-categorical colimits called codescent objects. We isolate the extra structure present on these simplicial objects which enable their codescent object…

Constructing cell data for diagram algebras

We show how the treatment of cellularity in families of algebras arising from diagram calculi, such as Jones' Temperley--Lieb wreaths, variants on Brauer's centralizer algebras, and the contour algebras of Cox et al (of which many algebras are special cases), may be unified using the theory of tabular algebras. This improves an earlier result of the first author (whose hypotheses covered only the Brauer algebra from among these families).

Applications of mutations in the derived categories of weighted projective lines to Lie and quantum algebras

Let $\rm{coh}\mathbb{X}$ be the category of coherent sheaves over a weighted projective line $\mathbb{X}$ and let $D^b(\rm{coh}\mathbb{X})$ be its bounded derived category. The present paper focuses on the study of the right and left mutation functors arising in $D^b(\rm{coh}\mathbb{X})$ attached to certain line bundles. As applications, we first show that these mutation functors give rise to simple reflections for the Weyl group of the star shaped quiver $Q$ associated with $\mathbb{X}$. By further dealing with the Ringel--Hall algebra of $\mathbb{X}$, we show that these functors provide a re…

源于二周期射影复形与导出范畴的李代数

Lie algebras arising from two-periodic projective complex and derived categories

本文利用二周期射影复形范畴及其稳定范畴，分别为有限维代数的模范畴构造了李代数，并证明了这两种构造在自然函子与 Grothendieck 群同构下诱导出李代数同构，从而统一了两种结构常数的计算。