首页/论文/math/0203157

arXiv math/02031572002-03-15math.QAmath.GT

Indecomposable racks of order p^2

Matias Graña

备注：: 8 pages

英文摘要：

We classify indecomposable racks of order p^2 (p a prime). There are 2p^2 - 2p - 2 isomorphism classes, among which 2p^2 - 3p - 1 correspond to quandles. In particular, we prove that an indecomposable quandle of order p^2 is affine (=Alexander). One of the results yielding this classification is the computation of the quandle nonabelian second cohomology group of an indecomposable quandle of prime order; which turns out to be trivial, as in the abelian case.

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。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

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

相关论文

On the classification of quandles of low order

Using the classification of transitive groups we classify indecomposable quandles of size <36. This classification is available in Rig, a GAP package for computations related to racks and quandles. As an application, the list of all indecomposable quandles of size <36 not of type D is computed.

2012-01-26math.GTmath.QA

An explicit description of the second cohomology group of a quandle

We use the inflation-restriction sequence and a result of Etingof and Graña on the rack cohomology to give a explicit description of 2-cocycles of finite indecomposable quandles with values in an abelian group. Several applications are given.

2016-09-06math.KTmath.QA

Enumeration of racks and quandles up to isomorphism

Racks and quandles are prominent set-theoretical solutions of the Yang-Baxter equation. We enumerate racks and quandles of orders $n\le 13$ up to isomorphism, improving upon the previously known results for $n\le 8$ and $n\le 9$, respectively. The enumeration is based on the classification of subgroups of small symmetric groups up to conjugation, on a representation of racks and quandles in symmetric groups due to Joyce and Blackburn, and on a number of theoretical and computational observations concerning the representation. We explicitly find representatives of isomorphism types of racks of …

2019-11-12math.QA

广义Legendrian racks的分类与结构

Classification and structure of generalized Legendrian racks

University of Pittsburgh（匹兹堡大学）

本文发展了用于研究Legendrian纽结的广义Legendrian racks的结构理论，通过群论刻画、穷举搜索算法分类、范畴等价构造及张量积研究，取得相关分类与结构成果。

2026-09-01math.GRmath.GTmath.QA

Yang-Baxter deformations of quandles and racks

Given a rack Q and a ring A, one can construct a Yang-Baxter operator c_Q: V tensor V --> V tensor V on the free A-module V = AQ by setting c_Q(x tensor y) = y tensor x^y for all x,y in Q. In answer to a question initiated by D.N. Yetter and P.J. Freyd, this article classifies formal deformations of c_Q in the space of Yang-Baxter operators. For the trivial rack, where x^y = x for all x,y, one has, of course, the classical setting of r-matrices and quantum groups. In the general case we introduce and calculate the cohomology theory that classifies infinitesimal deformations of c_Q. In many cas…

2005-06-30math.GTmath.QA

共轭子广群的良好对合

Good involutions of conjugation subquandles

该研究针对带良好对合的广群分类难题，分析共轭子广群等的良好对合并得到数量界，实现相关计算算法、构造非对合对称广群族，完成低阶相关结构分类并证明范畴等价关系。

2025-08-04math.GRmath.GTmath.QA

更多相关论文收起更多论文

On the lattice of subracks of the rack of a finite group

In this paper we initiate the study of racks from the combined perspective of combinatorics and finite group theory. A rack R is a set with a self-distributive binary operation. We study the combinatorics of the partially ordered set {\cal R}(R) of all subracks of R with inclusion as the order relation. Groups G with the conjugation operation provide an important class of racks. For the case R = G we show that -> the order complex of {\cal R}(R) has the homotopy type of a sphere, -> the isomorphism type of {\cal R}(R) determines if G is abelian, nilpotent, supersolvable, solvable or simple, ->…

2015-12-04math.COmath.GRmath.QA

虚拟quandle的枚举

Enumeration of virtual quandles up to isomorphism

本文通过计算机搜索对虚拟quandle和racks进行到同构的分类，并计算了有限循环群holomorphs的类数。

2025-06-19math.GRmath.GTmath.QA

$f$-Racks, $f$-Quandles, their Extensions and Cohomology

The purpose of this paper is to introduce and study the notions of $f$-rack and $f$-quandle which are obtained by twisting the usual equational identities by a map. We provide some key constructions, examples and classification of low order $f$-quandles. Moreover, we define modules over $f$-racks, discuss extensions and define a cohomology complex for $f$-quandles.

2016-11-29math.ATmath.QAmath.RA

Indecomposable set-theoretical solutions to the Quantum Yang-Baxter Equation on a set with prime number of elements

In this paper we show that all indecomposable nondegenerate set-theoretical solutions to the Quantum Yang-Baxter equation on a set of prime order are affine, which allows us to give a complete and very simple classification of such solutions. This result is a natural application of the general theory of set-theoretical solutions to the quantum Yang-Baxter equation. It is also a generalization of the corresponding statement for involutive set-theoretical solutions proved in an earlier paper of P.E. and A.S. with T.Schedler. In order to prove our main result, we use the classification theory dev…

2000-07-26math.QA

Indecomposable solutions of the Yang-Baxter equation of square-free cardinality

Indecomposable involutive non-degenerate set-theoretic solutions $(X,r)$ of the Yang-Baxter equation of cardinality $p_1\cdots p_n$, for different prime numbers $p_1,\ldots, p_n$, are studied. It is proved that they are multipermutation solutions of level $\leq n$. In particular, there is no simple solution of a non-prime square-free cardinality. This solves a problem stated in [F. Cedó, J. Okniński, Constructing finite simple solutions of the Yang-Baxter equation, Adv. Math. 391 (2021), 107968] and provides a far reaching extension of several earlier results on indecomposability of solutions.…

2022-12-13math.QA