A braid group representation derived from handlebodies

递减积的导出辫及其范畴

Derived Braids of Decreasing Products and their Categories

本文证明由偶数股分量辫构成的递减积具有等价导出辫，从而将导出辫相等推广为Yang-Baxter方程的广义形式，并关联Joyal-Street相干性结果。

A rigidity result for extensions of braided tensor C*-categories derived from compact matrix quantum groups

Let G be a classical compact Lie group and G_μthe associated compact matrix quantum group deformed by a positive parameter μ(or a nonzero and real μin the type A case). It is well known that the category Rep(G_μ) of unitary f.d. representations of G_μis a braided tensor C*-category. We show that any braided tensor *-functor from Rep(G_μ) to another braided tensor C*-category with irreducible tensor unit is full if |μ|\neq 1. In particular, the functor of restriction to the representation category of a proper compact quantum subgroup, cannot be made into a braided functor. Our result also shows…

Support varieties over skew complete intersections via derived braided Hochschild cohomology

In this article we study a theory of support varieties over a skew complete intersection $R$, i.e. a skew polynomial ring modulo an ideal generated by a sequence of regular normal elements. We compute the derived braided Hochschild cohomology of $R$ relative to the skew polynomial ring and show its action on $\mathrm{Ext}_R(M,N)$ is noetherian for finitely generated $R$-modules $M$ and $N$ respecting the braiding of $R$. When the parameters defining the skew polynomial ring are roots of unity we use this action to define a support theory. In this setting applications include a proof of the Gen…

Link invariants from finite categorical groups and braided crossed modules

We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, 2-crossed modules and braided crossed modules. We prove that our construction includes all rack and quandle cohomology (framed) link invariants, as well as the Eisermann invariant of knots, for which we also find a lifting by using braided crossed modules.

Sutured Khovanov homology, Hochschild homology, and the Ozsvath-Szabo spectral sequence

In 2001, Khovanov and Seidel constructed a faithful action of the (m+1)-strand braid group on the derived category of left modules over a quiver algebra, A_m. We interpret the Hochschild homology of the Khovanov-Seidel braid invariant as a direct summand of the sutured Khovanov homology of the annular braid closure.

Braid cobordisms, triangulated categories, and flag varieties

We argue that various braid group actions on triangulated categories should be extended to projective actions of the category of braid cobordisms and illustrate how this works in examples. We also construct actions of both the affine braid group and the braid cobordism category on the derived category of coherent sheaves on the cotangent bundle to the full flag variety.

Stability conditions and the braid group

We find stability conditions ([Do], [Br]) on some derived categories of differential graded modules over a graded algebra studied in [RZ], [KS]. This category arises in both derived Fukaya categories and derived categories of coherent sheaves. This gives the first examples of stability conditions on the A-model side of mirror symmetry, where the triangulated category is not naturally the derived category of an abelian category. The existence of stability conditions, however, gives many such abelian categories, as predicted by mirror symmetry. In our examples in 2 dimensions we completely descr…

Quivers, Floer cohomology, and braid group actions

We consider the derived categories of modules over a certain family A_m of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of type A_m. We show that each of these two rather different objects encodes the topology of curves on an (m+1)-punctured disc. We prove that the braid group B_{m+1} acts faithfully on the derived category of A_m-modules and that it injects into the symplectic mapping class group of Milnor fibers. The philosophy behind our results is as follows. Using Floer cohomology, one should…

Braid groups and Kleinian singularities

We establish faithfulness of braid group actions generated by twists along an ADE configuration of $2$-spherical objects in a derived category. Our major tool is the Garside structure on braid groups of type ADE. This faithfulness result provides the missing ingredient in Bridgeland's description of a space of stability conditions associated to a Kleinian singularity.

Braided Hochschild cohomology and Hopf actions

We show that the braided Hochschild cohomology, of an algebra in a suitably algebraic braided monoidal category, admits a graded ring structure under which it is braided commutative. We then give a canonical identification between the usual Hochschild cohomology ring of a smash product and the (derived) invariants of its braided Hochschild cohomology ring. We apply our results to identify the associative formal deformation theory of a smash product with its formal deformation theory as a module algebra over the given Hopf algebra (when the Hopf algebra is sufficiently semisimple). As a second …

Categorification of infinite-dimensional $\mathfrak{sl}_2$-modules and braid group 2-actions I: tensor products

This is the first part of a series of two papers aiming to construct a categorification of the braiding on tensor products of Verma modules, and in particular of the Lawrence--Krammer--Bigelow representations. \\ In this part, we categorify all tensor products of Verma modules and integrable modules for quantum $\mathfrak{sl_2}$. The categorification is given by derived categories of dg versions of KLRW algebras which generalize both the tensor product algebras of Webster, and the dg-algebras used by Lacabanne, the second author and Vaz. We compute a basis for these dgKLRW algebras by using re…

Representations of the Loop Braid Group and Aharonov-Bohm like effects in discrete (3+1)-dimensional higher gauge theory

We show that representations of the loop braid group arise from Aharonov-Bohm like effects in finite 2-group (3+1)-dimensional topological higher gauge theory. For this we introduce a minimal categorification of biracks, which we call W-bikoids (welded bikoids). Our main example of W-bikoids arises from finite 2-groups, realised as crossed modules of groups. Given a W-bikoid, and hence a groupoid of symmetries, we construct a family of unitary representations of the loop braid group derived from representations of the groupoid algebra. We thus give a candidate for higher Bais' flux metamorphos…

Braid group approach to the derivation of universal Ř matrices

A new method for deriving universal Ř matrices from braid group representation is discussed. In this case, universal Ř operators can be defined and expressed in terms of products of braid group generators. The advantage of this method is that matrix elements of Ř are rank independent, and leaves multiplicity problem concerning coproducts of the corresponding quantum groups untouched. As examples, Ř matrix elements of $[1]\times [1]$, $[2]\times [2]$, $[1^{2}]\times [1^{2}]$, and $[21]\times [21]$ with multiplicity two for $A_{n}$, and $[1]\times [1]$ for $B_{n}$, $C_{n}$, and $D_{n}$ type quan…

Quasitriangular Hopf algebras, braid groups and quantum entanglement

The aim of the paper is to provide an method to obtain representations of the braid group through a set of quasitriangular Hopf algebras. In particular, these algebras may be derived from group algebras of cyclic groups with additional algebraic structures. In this context, by using the flip operator, it is possible to construct R-matrices that can be regarded as quantum logic gates capable of preserving quantum entanglement.

柄体群表示的相容系统的分类

Classification of Consistent Systems of Handlebody Group Representations

本研究围绕柄体群表示的相容系统分类问题，以柄体模operad为核心工具，证明其在对称幺半双范畴上的代数（ansular函子）等价于自对偶平衡辫子代数，线性框架下对应ribbon Grothendieck-Verdier范畴，还给出柄体对应向量空间的公式并可应用于顶点算子代数。

量子簇代数的拟同态

Quasi-homomorphisms of quantum cluster algebras

本文研究量子簇代数的拟同态，证明辫群生成元保持量子格拉斯曼簇代数的拟交换与交换关系，并在猜想下成为拟自同构，使辫群作用于该代数。

非组合对合辫：量子代数 $\mathfrak{gl}_{k,m}$

Non-combinatorial involutive braidings: the quantum algebra $\mathfrak{gl}_{k,m}$

研究辫方程的非组合对合解，将其视为置换映射的特殊形变，并引入关联的量子代数 $\mathfrak{gl}_{k,m}$ Yangian，构造其最高权模并建立与杨表形状的联系。

辫子交换代数的同伦不变量与有限张量范畴的德利涅猜想

Homotopy Invariants of Braided Commutative Algebras and the Deligne Conjecture for Finite Tensor Categories

本研究利用辫子Operad理论证明有限张量范畴中两类辫子交换代数的同伦不变量具有微分分次$E_2$-代数结构，明确了其霍赫希尔德上链复形德利涅$E_2$-结构的诱导来源，提升了Ext代数的括号结构并改进了相关包含映射的结果。

来自环辫群的 Hecke 代数的推广

Generalisations of Hecke algebras from Loop Braid Groups

本文基于环辫群和 Burau 表示的扩张构造了 Hecke 代数的推广 $LH_n$，证明其具有有限维性、包含关系、半单商由 Pascal 三角形描述、A 型箭图等表示论性质，并分析了其与参数 $t$ 及商代数 $SP_n$ 的同构关系。

Crossingless matchings and the cohomology of (n,n) Springer varieties

The sequence of rings $H^n, n\ge 0,$ introduced in math.QA/0103190, controls categorification of the quantum sl(2) invariant of tangles. We prove that the center of $H^n$ is isomorphic to the cohomology ring of the (n,n) Springer variety and show that the braid group action in the derived category of $H^n$-modules descends to the Springer action of the symmetric group.

Regular basis and R-matrices for the su(n)_k Knizhnik-Zamolodchikov equation

Dynamical R-matrix relations are derived for the group-valued chiral vertex operators in the SU(n) WZNW model from the KZ equation for a general four-point function including two step operators. They fit the exchange relations for the U_q(sl_n) covariant quantum matrix derived previously by solving the dynamical Yang-Baxter equation. As a byproduct, we extend the regular basis introduced earlier for SU(2) chiral fields to SU(n) step operators and display the corresponding triangular matrix representation of the braid group.

Quantum Divided Power Algebra, q-Derivatives and Some New Quantum Groups

The discussions in the present paper arise from exploring intrinsically the structure nature of the quantum $n$-space. A kind of braided category $\Cal {GB}$ of $\La$-graded $þ$-commutative associative algebras over a field $k$ is established. The quantum divided power algebra over $k$ related to the quantum $n$-space is introduced and described as a braided Hopf algebra in $\Cal {GB}$ (in terms of its 2-cocycle structure), over which the so called special $q$-derivatives are defined so that several new interesting quantum groups, especially, the quantized polynomial algebra in $n$ variables (…

The Twisted Heisenberg Algebra ${\cal U}_{h,w}({\cal H}(4))$

A two parametric deformation of the enveloping Heisenberg algebra ${\cal H}(4)$ which appear as a combination of the standard and a nonstandard quantization given by Ballesteros and Herranz is defined and proved to be Ribbon Hopf algebra. The universal ${\cal R}$-matrix and its associated quantum group are constructed. New solution of Braid group are obtained. The contribution of these parameters in invariants of links and WZW model are analyzed. General results for twisted Ribbon Hopf algebra are derived.

环路空间的 Morse 理论与 Hecke 代数

Morse theory of loop spaces and Hecke algebras

本文通过定义基于基点组的多环路 $A_\infty$-代数，将 Morse 理论与辫子 skein 代数及 Hecke 代数联系起来，并证明其与环绕高维 Heegaard Floer $A_\infty$-代数等价，同时计算了 $S^2$ 情形的导出 Hecke 代数。

Set-theoretic solutions of the Yang-Baxter equation, Braces, and Symmetric groups

We involve simultaneously the theory of matched pairs of groups and the theory of braces to study set-theoretic solutions of the Yang-Baxter equation (YBE). We show the intimate relation between the notions of a symmetric group (a braided involutive group) and a left brace, and find new results on symmetric groups of finite multipermutation level and the corresponding braces. We introduce a new invariant of a symmetric group $(G,r)$, \emph{the derived chain of ideals of} $G$, which gives a precise information about the recursive process of retraction of $G$. We prove that every symmetric group…

A category for the adjoint representation

We construct an abelian category C and exact functors in C which on the Grothendieck group descend to the action of a simply-laced quantum group in its adjoint representation. The braid group action in the adjoint representation lifts to an action in the derived category of C. The category C is the direct sum of a semisimple category and the category of modules over a certain algebra A, associated to a Dynkin diagram. In the second half of the paper we show how these algebras appear in the modular representation theory and in the McKay correspondence and explore their relationship with root sy…