Faithful action of braid group on bosonic extensions
Masaki Kashiwara, Myungho Kim, Se-jin Oh, Euiyong Park
本研究证明了辫群在量子群玻色扩展上的作用是忠实的，扩展了Lusztig的对称性概念。
辫群作用于量子群的玻色扩展已在近期工作中引入，可以视为Lusztig对量子群的对称性的推广。在本笔记中，我们证明了这一辫群作用的忠实性。
The braid group action on the bosonic extension of the quantum group has been introduced in recent works, and it can be regarded as a generalization of Lusztig's symmetries on the quantum group. In this notes, we prove the faithfulness of this braid group action.
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Braid symmetries on bosonic extensions
本研究在任意型玻色子扩张上引入满足辫子关系的自同构族，利用其辫子群作用定义具PBW型基的正辫子词子代数，并给出正玻色子扩张张量积分解的应用。
Faithfulness of the 2-Braid Group via Zigzag Algebra in Type B
本文通过证明B型之字形代数上的双模范畴是B型Soergel双模范畴的商范畴，为Rouquier关于B型2-辫子群忠实性的猜想提供了另一种证明。
We establish strong constraints on the kernel of the (reduced) Burau representation $β_4:B_4\to \text{GL}_3\left(\mathbb{Z}\left[q^{\pm 1}\right]\right)$ of the braid group $B_4$. We develop a theory to explicitly determine the entries of the Burau matrices of braids in $B_4$, and this is an important step toward demonstrating that $β_4$ is faithful (a longstanding question posed in the 1930s). The theory is based on a novel combinatorial interpretation of $β_4\left(g\right)$, in terms of the Garside normal form of $g\in B_4$ and a new product decomposition of positive braids. We develop cance…
We prove that the action of a generalized braid group on an enhanced triangulated categories, generated by spherical twist functors along an ADE-configuration of $ω$-spherical objects, is faithful for any integer $ω\neq 1$.
Bicomplexes of vector spaces frequently appear throughout algebra and geometry. In Section 2 we explain how to think about the arrows in the spectral sequence of a bicomplex via its indecomposable summands. Polycomplexes seem to be much more rare. In Section 3 of this paper we rethink a well-known faithful categorical braid group action via an action on the stable category of tricomplexes.
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.
We construct a faithful tensor representation for the Yokonuma-Hecke algebra Y, and use it to give a concrete isomorphism between Y and Shoji's modified Ariki-Koike algebra. We give a cellular basis for Y and show that the Jucys-Murphy elements for Y are JM-elements in the abstract sense. Finally, we construct a cellular basis for the Aicardi-Juyumaya algebra of braids and ties.
We prove that any `finite-type' component of a stability space of a triangulated category is contractible. The motivating example of such a component is the stability space of the Calabi--Yau-$N$ category $\mathcal{D}(Γ_N Q)$ associated to an ADE Dynkin quiver. In addition to showing that this is contractible we prove that the braid group $\operatorname{Br}(Q)$ acts freely upon it by spherical twists, in particular that the spherical twist group $\operatorname{Br}(Γ_N Q)$ is isomorphic to $\operatorname{Br}(Q)$. This generalises Brav-Thomas' result for the $N=2$ case. Other classes of triangul…
We define a family of the braid group representations via the action of the $R$-matrix (of the quasitriangular extension) of the restricted quantum $\mathfrak{sl}(2)$ on a tensor power of a simple projective module. This family is an extension of the Lawrence representation specialized at roots of unity. Although the center of the braid group has finite order on the specialized Laurence representations, this action is faithful for our extension.
Using a quiver algebra of a cyclic quiver, we construct a faithful categorical action of the extended braid group of affine type A on its bounded homotopy category of finitely generated projective modules. The algebra is trigraded and we identify the trigraded dimensions of the space of morphisms of this category with intersection numbers coming from the topological origin of the group.
We construct a finite dimensional quiver algebra from the non-simply laced type $B$ Dynkin diagram, which we call the type $B$ zigzag algebra. This leads to a faithful categorical action of the type $B$ braid group $\mathcal{A}(B)$, acting on the homotopy category of its projective modules. This categorical action is also closely related to the topological action of $\mathcal{A}(B)$, viewed as mapping class group of the punctured disc -- hence our exposition can be seen as a type $B$ analogue of Khovanov-Seidel's work in arXiv:math/0006056v2. Moreover, we show that certain category of bimodule…
Extriangulated structures from stability conditions and application to braid representations
本文通过Bridgeland稳定性条件赋予三角范畴外三角结构，并揭示了Burau表示与Lawrence-Krammer-Bigelow表示之间的关联。
Let B be the generalized braid group associated to some finite complex reflection group. We define a representation of B of dimension the number of reflections of the corresponding reflection group, which generalizes the Krammer representation of the classical braid groups, and is thus a good candidate in view of proving the linearity of these groups. We decompose this representation in irreducible components and compute its Zariski closure, as well as its restriction to parabolic subgroups. We prove that it is faithful when W is a Coxeter group of type ADE and odd dihedral types, and conjectu…
In this paper we describe the structure of a group of conjugating automorphisms $C_n$ of free group and prove that this structure is similar to the structure of a braid group $B_n$ with $n>1$ strings. We find the linear representation of group $C_n$. Also we prove that the braid group $B_n(S^2)$ of 2--sphere, mapping class group M(0,n) of the $n$--punctured 2--sphere and the braid group $B_3(P^2)$ of the projective plane are linear. Using result of J. Dyer, E. Formanek, E. Grossman and the faithful linear representation of Lawrence--Krammer of $B_4$ we construct faithful linear representation …
We construct two families of representations of the braid group $B_n$ by considering conjugation actions on congruence subgroups of $GL_{n-1}(Z[t^{\pm 1},q^{\pm 1}])$. We show that many of these representations are faithful modulo the center of $B_n$.
We investigate the relation between the Garside normal form for positive braids and the $2$-braid group defined by Rouquier. Inspired by work of Brav and Thomas we show that the Garside normal form is encoded in the action of the $2$-braid group on a certain categorified left cell module. This allows us to deduce the faithfulness of the $2$-braid group in finite type. We also give a new proof of Paris' theorem that the canonical map from the generalized braid monoid to its braid group is injective in arbitrary type.
We consider the algebra ${\cal E}_n(u)$ introduced by F. Aicardi and J. Juyumaya as an abstraction of the Yokonuma-Hecke algebra. We construct a tensor space representation for ${\cal E}_n(u)$ and show that this is faithful. We use it to give a basis for ${\cal E}_n(u)$ and to classify its irreducible representations.
We construct an action of a braid group associated to a complete graph on the derived category of a certain symmetric Nakayama algebra which is also a Brauer star algebra with no exceptional vertex. We connect this action with the affine braid group action on Brauer star algebras defined by Schaps and Zakay-Illouz. We show that for Brauer star algebras with no exceptional vertex, the action is faithful.
On the Faithfulness of a Family of Representations of the Singular Braid Monoid $SM_n$
本文研究奇异辫子幺半群表示族 $Φ_{a,b,c}$ 的忠实性，给出单参数族不忠实的充要条件，分析 $n=2$ 时核的性质与平凡核情形，并刻画 $n\geq3$ 时核元素形态。
Let $C_n$ be the group of conjugating automorphisms. We study the representation $ρ$ of $C_n$, an extension of Lawrence-Krammer representation of the braid group $B_n$, defined by Valerij G. Bardakov. As Bardakov proved that the representation $ρ$ is unfaithful for $n \geq 5$, the cases $n=3,4$ remain open. In our work, we make attempts towards the faithfulness of $ρ$ in the case $n=3$.
On cluster structures of bosonic extensions
本文研究了量子单侧坐标环的玻色扩展上的量子聚类结构，通过分析Lusztig参数化和辫子移动下的过渡映射，证明了所得的量子聚类结构与所选表达式无关，并验证了Kashiwara-Kim-Oh-Park猜想，同时建立了广义量子 minors 的量子 T-系统关系。
Monoidal Categorification and Quantization of Braid Varieties
针对单链型单连通代数群的正辫子字，本文构造辫子簇量子簇代数的幺范畴化，将其与局域化量子环等同，验证特殊化同态恢复原簇代数，还建立了双弦簇变量的量子网格子式与广义量子T-系统。
In this paper we propose, firstly, a categorification of virtual braid groups and groupoids in terms of "locally" braided objects in a symmetric category (SC), and, secondly, a definition of self-distributive structures (SDS) in an arbitrary SC. SDS are shown to produce braided objects in a SC. As for examples, we interpret associativity and the Jacobi identity in a SC as generalized self-distributivity, thus endowing associative and Leibniz algebras with a "local" braiding. Our "double braiding" approach provides a natural interpretation for virtual racks and twisted Burau representation. Hom…
Some remarks about the faithfulness of the Burau representation of Artin--Tits groups
本文将Burau表示的忠实性问题推广到Artin–Tits群，并用推广Bigelow曲线策略的范畴算法证明其在仿射型$\ ilde{A_3}$及型$D_4$的若干有限环上不忠实。
A connection is made between the Krammer representation and the Birman-Murakami-Wenzl algebra. Inspired by a dimension argument, a basis is found for a certain irrep of the algebra, and relations which generate the matrices are found. Following a rescaling and change of parameters, the matrices are found to be identical to those of the Krammer representation. The two representations are thus the same, proving the irreducibility of one and the faithfulness of the other.
Faithful Burau-like representations of some rank two Garside groups and torus knot groups
本文提出一种方法，用于生成G(n,m)群在GL₂(C[t±¹, q±¹])中的忠实表示，其中G(n,m)是Garside群，其元素的Garside正规形式可通过矩阵显式恢复。该方法构造了多个忠实表示，包括推广到G(n,m)类群的Burau表示，这些群是扭结群。
We construct a faithful categorical action of the type $B$ braid group on the bounded homotopy category of finitely generated projective modules over a finite dimensional algebra which we call the type $B$ zigzag algebra. This categorical action is closely related to the action of the type $B$ braid group on curves on the disc. Thus, our exposition can be seen as a type $B$ analogue of the work of Khovanov-Seidel in arXiv:math/0006056. Moreover, we relate our topological (respectively categorical) action of the type $B$ Artin braid group to their topological (respectively categorical) action o…
In the present paper we study structural aspects of certain quotients of braid groups and virtual braid groups. In particular, we construct and study linear representations $B_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}]\right)$, $VB_n\to {\rm GL}_{n(n-1)/2}\left(\mathbb{Z}[t^{\pm1}, t_1^{\pm1},t_2^{\pm1},\ldots, t_{n-1}^{\pm1}]\right)$ which are connected with the famous Lawrence-Bigelow-Krammer representation. It turns out that these representations are faithful representations of crystallographic groups $B_n/P_n'$, $VB_n/VP_n'$, respectively. Using these representations we study certa…
We study Artin-Tits braid groups $\mathbb{B}_W$ of type ADE via the action of $\mathbb{B}_W$ on the homotopy category $\mathcal{K}$ of graded projective zigzag modules (which categorifies the action of the Weyl group $W$ on the root lattice). Following Brav-Thomas, we define a metric on $\mathbb{B}_W$ induced by the canonical $t$-structure on $\mathcal{K}$, and prove that this metric on $\mathbb{B}_W$ agrees with the word-length metric in the canonical generators of the standard positive monoid $\mathbb{B}_W^+$ of the braid group. We also define, for each choice of a Coxeter element $c$ in $W$…
We study the plactic algebra and its action on bosonic particle configurations in the classical case. These particle configurations together with the action of the plactic generators can be identified with crystals of the quantum analogue of the symmetric tensor representations in type A. It turns out that this action factors over a quotient algebra that we call partic algebra, whose induced action on bosonic particle configurations is faithful. We describe a basis of the partic algebra explicitly in terms of a normal form for monomials, and we compute the center of the partic algebra.
We define geodesic normal forms for the general series of complex reflection groups G(de,e,n). This requires the elaboration of a combinatorial technique in order to determine minimal word representatives and to compute the length of the elements of G(de,e,n) over some generating set. Using these geodesic normal forms, we construct intervals in G(e,e,n) that give rise to Garside groups. Some of these groups correspond to the complex braid group B(e,e,n). For the other Garside groups that appear, we study some of their properties and compute their second integral homology groups. Inspired by th…
The faithfulness of an extension of Lawrence-Krammer representation on the group of conjugating automorphisms $C_n$ in the cases $n=3$ and $n=4$
本研究针对共轭自同构群$C_n$上Lawrence-Krammer表示的扩张$\rho$的忠实性问题，填补了$n=3$、$n=4$情形的研究空白，证明这两种情形下该表示均不忠实。
Local Representations of the Flat Virtual Braid Group
本文分类了平坦虚拟辫子群FVB_2到GL_2(C)的复局部表示，并研究了其不可约性、忠实性条件，同时推广到n≥3和n+1维情形。
Torus actions on compactified braid varieties and polytopality of subword complexes
本文证明简单李群辫子簇的簇膨胀作用可延拓至砖紧化，并由此得出闭 Richardson 簇的环面忠实作用及辫子簇为环面的充要条件，进而通过双根自由词为子词复形的多面体性提供新方法并分类相关 Bruhat 区间。
We study fundamental group of the exchange graphs for the bounded derived category D(Q) of a Dynkin quiver Q and the finite-dimensional derived category D(Γ_N Q) of the Calabi-Yau-N Ginzburg algebra associated to Q. In the case of D(Q), we prove that its space of stability conditions (in the sense of Bridgeland) is simply connected; as applications, we show that its Donanldson-Thomas invariant can be calculated via a quantum dilogarithm function on exchange graphs. In the case of D(Γ_N Q), we show that faithfulness of the Seidel-Thomas braid group action (which is known for Q of type A or N = …
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