From racks to pointed Hopf algebras

Andruskiewitsch, Nicolás; Graña, M.

doi:10.1016/s0001-8708(02)00071-3

Cited by 290 publications

(489 citation statements)

References 26 publications

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“…As in Subsection 1.2, many results originally established using quandles have been subsequently extended to arbitrary racks [2]. Also, racks proved to play a fundamental rôle in the classification of finite-dimensional pointed Hopf algebras [1]. The question of whether one could go one step further and work with more general LD-systems, specifically with Laver tables, remains open.…”

Section: The Case Of the Laver Tablesmentioning

confidence: 99%

Laver’s results and low-dimensional topology

Dehornoy

2015

Arch. Math. Logic

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Abstract. In connection with his interest in selfdistributive algebra, Richard Laver established two deep results with potential applications in low-dimensional topology, namely the existence of what is now known as the Laver tables and the well-foundedness of the standard ordering of positive braids. Here we present these results and discuss the way they could be used in topological applications.Richard Laver established two remarkable results that might lead to significant applications in low-dimensional topology, namely the existence of a series of finite structures satisfying the left-selfdistributive law, now known as the Laver tables, and the well-foundedness of the standard ordering of Artin's positive braids. In this text, we shall explain the precise meaning of these results and discuss their (past or future) applications in topology. In one word, the current situation is that, although the depth of Laver's results is not questionable, few topological applications have been found. However, the example of braid groups orderability shows that, once initial obstructions are solved, topological applications of algebraic results involving selfdistributivity can be found; the situation with Laver tables is presumably similar, and the only reason explaining why so few applications are known is that no serious attempt has been made so far, mainly because the results themselves remain widely unknown in the topology community.Therefore this text is more a program than a report on existing results. Our aim is to provide a self-contained and accessible introduction to the subject, hopefully helping the algebraic and topological communities to better communicate. Most of the results mentioned below have already appeared in literature, a number of them even belonging to the folklore of their domain (whereas ignored outside of it). However, at least the observations about cocycles for Laver tables mentioned in Subsection 1.3 (and established in another paper) are new.Very naturally, the text comprises two sections, one devoted to Laver tables, and one devoted to the well-foundedness of the braid ordering. It should be noted that the above two topics (Laver tables, well-foundedness of the braid ordering) do not exhaust Laver's contributions to selfdistributive algebra and, from there, to potential topological applications: in particular, Laver constructed powerful tools for investigating free LD-structures, leading to applications of their own [68,69,70]. However, connections with topology are less obvious in these cases and we shall not develop them here (see the other articles in this volume).

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Section: The Case Of the Laver Tablesmentioning

confidence: 99%

Laver’s results and low-dimensional topology

Dehornoy

2015

Arch. Math. Logic

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“…Consider the affine rack X = (F 5 , 2) and the constant cocycle q ≡ −1. The Nichols algebra B(X, q), computed in [AG1], has dimension 1280 and can be presented by generators x 0 , . .…”

Section: Definition 410 a Good Module Of Relations Is A Graded Yetter-mentioning

confidence: 99%

Lifting via cocycle deformation

Andruskiewitsch

Angiono

Iglesias

et al. 2014

Journal of Pure and Applied Algebra

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“…Many variations of this idea exist, including the quandle homology described in [3], the twisted quandle homology described in [4], the quandle homology with coefficients in a quandle module ( [1], [10]), the biquandle version known as YangBaxter homology ( [5]), and a unifying approach to the homology of quandles and Lie algebras ( [2]). …”

Section: Yang-baxter Cohomologymentioning

confidence: 99%

Virtual Yang-Baxter cocycle invariants

Ceniceros

Nelson

2009

Trans. Amer. Math. Soc.

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Abstract. We extend the Yang-Baxter cocycle invariants for virtual knots by augmenting Yang-Baxter 2-cocycles with cocycles from a cohomology theory associated to a virtual biquandle structure. These invariants coincide with the classical Yang-Baxter cocycle invariants for classical knots but provide extra information about virtual knots and links. In particular, they provide a method for detecting non-classicality of virtual knots and links.

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From racks to pointed Hopf algebras

Cited by 290 publications

References 26 publications

Laver’s results and low-dimensional topology

Laver’s results and low-dimensional topology

Lifting via cocycle deformation

Virtual Yang-Baxter cocycle invariants

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