Garside and locally Garside categories
英文摘要:
We define and give axioms for Garside and locally Garside categories. We give an application to Coxeter and Artin groups and Deligne-Lusztig varieties.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
François Digne, Jean Michel
We define and give axioms for Garside and locally Garside categories. We give an application to Coxeter and Artin groups and Deligne-Lusztig varieties.
机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。
继续探索
In connection with the emerging theory of Garside categories, we develop the notions of a left-Garside category and of a locally left-Garside monoid. In this framework, the connection between the self-distributivity law LD and braids amounts to the result that a certain category associated with LD is a left-Garside category, which projects onto the standard Garside category of braids. This approach leads to a realistic program for establishing the Embedding Conjecture of [Dehornoy, Braids and Self-distributivity, Birkhauser (2000), Chap. IX].
Garside groupoids, as recently introduced by Krammer, generalise Garside groups. A weak Garside group is a group that is equivalent as a category to a Garside groupoid. We show that any periodic loop in a Garside groupoid $\CG$ may be viewed as a Garside element for a certain Garside structure on another Garside groupoid $\CG_m$, which is equivalent as a category to $\CG$. As a consequence, the centraliser of a periodic element in a weak Garside group is a weak Garside group. Our main tool is the notion of divided Garside categories, an analog for Garside categories of Bökstedt-Hsiang-Madsen's…
We describe how an Ore category with a Garside family can be used to construct a classifying space for its fundamental group(s). The construction simultaneously generalizes Brady's classifying space for braid groups and the Stein--Farley complexes used for various relatives of Thompson's groups. It recovers the fact that Garside groups have finite classifying spaces. We describe the categories and Garside structures underlying certain Thompson groups. The Zappa--Szép product of categories is introduced and used to construct new categories and groups from known ones. As an illustration of our m…
Generalization of the Dehornoy-Lafont order complex to categories. Application to exceptional braid groups
本文将Dehornoy-Lafont序复形推广到Garside范畴并提出降时计算技术,用以补全例外复辫子群的同调结果,重点研究$B(G_{31})$。
We describe new types of normal forms for braid monoids, Artin-Tits monoids, and, more generally, for all monoids in which divisibility has some convenient lattice properties (``locally Garside monoids''). We show that, in the case of braids, one of these normal forms coincides with the normal form introduced by Burckel and deduce that the latter can be computed easily. This approach leads to a new, simple description for the standard ordering (``Dehornoy order'') of Bn in terms of that of B(n-1), and to a quadratic upper bound for the complexity of this ordering.
We initiate the study of C*-algebras and groupoids arising from left regular representations of Garside categories, a notion which originated from the study of Braid groups. Every higher rank graph is a Garside category in a natural way. We develop a general classification result for closed invariant subspaces of our groupoids as well as criteria for topological freeness and local contractiveness, properties which are relevant for the structure of the corresponding C*-algebras. Our results provide a conceptual explanation for previous results on gauge-invariant ideals of higher rank graph C*-a…
Springer categories for regular centralizers in well-generated complex braid groups
本文研究了复杂辫群中正则中心化子的弹簧类别,通过Garside结构证明了关于中心的定理,并提供了Hurwitz-like的呈现方法。
Left regular representations of Garside categories II. Finiteness properties of topological full groups
本文研究与Garside范畴左正则表示群胚模型相关联的拓扑全群,发现底层小范畴的Garside结构可诱导双截面范畴的新Garside结构,使拓扑全群等同于其包络群胚的基本群,从而解决字问题并证明有限型移位乘积的拓扑全群为${\rm F}_{\infty}$型。
Simplicial pseudohyperplane arrangements give weak Garside groups
本文通过结合Bessis的Garside范畴与Salvetti的度量半球复形,证明了伪超平面排列复化补集及定向拟阵的Salvetti复形基本群均为弱Garside群,提供了该群的新实例。
Limits over orbit categories of locally finite groups
本文修正了先前论文中的错误,并探讨了局部有限群轨道范畴上逆极限的导出函子,以及有限群扩展的更高极限谱序列的推广。
Fix (not necessarily distinct) objects $i$ and $j$ of a locally small category $S$, and write $S_{ij}$ for the set of all morphisms $i\to j$. Fix a morphism $a\in S_{ji}$, and define an operation $\star_a$ on $S_{ij}$ by $x\star_ay=xay$ for all $x,y\in S_{ij}$. Then $(S_{ij},\star_a)$ is a semigroup, known as a sandwich semigroup, and denoted by $S_{ij}^a$. This article develops a general theory of sandwich semigroups in locally small categories. We begin with structural issues such as regularity, Green's relations and stability, focusing on the relationships between these properties on $S_{ij…
Fix sets $X$ and $Y$, and write $\mathcal{PT}_{XY}$ for the set of all partial functions $X\to Y$. Fix a partial function $a:Y\to X$, and define the operation $\star_a$ on $\mathcal{PT}_{XY}$ by $f\star_ag=fag$ for $f,g\in\mathcal{PT}_{XY}$. The sandwich semigroup $(\mathcal{PT}_{XY},\star_a)$ is denoted $\mathcal{PT}_{XY}^a$. We apply general results from Part I to thoroughly describe the structural and combinatorial properties of $\mathcal{PT}_{XY}^a$, as well as its regular and idempotent-generated subsemigroups, Reg$(\mathcal{PT}_{XY}^a)$ and $\mathbb E(\mathcal{PT}_{XY}^a)$. After describ…
扫码添加微信好友,提出您的宝贵建议 👇
💡 备注请填写:网站反馈
发表机构由 AI 从论文文本抽取,如发现错误,欢迎反馈修正建议。