arXiv subjects

Marco Bonatto

Publications and source records attributed to Marco Bonatto.

26 records · Page 2Linked to original sources

arXiv2020

Commutator theory for racks and quandles

We adapt the commutator theory of universal algebra to the particular setting of racks and quandles, exploiting a Galois connection between congruences and certain normal subgroups of the displacement group. Congruence properties such as abelianness and centrality are reflected by the corresponding relative displacement groups, and so do the global properties, solvability and nilpotence. To show the new tool in action, we present three applications: non-existence theorems for quandles (no connected involutory quandles of order $2^k$, no latin quandles of order $\equiv2\pmod4$), a non-colorability theorem (knots with trivial Alexander polynomial are not colorable by latin quandles), and a strengthening of Glauberman's results on Bruck loops of odd order.

math.GR↗

arXiv2020

Connected quandles of size $pq$ and $4p$

We classify all non-simple connected quandles of size $pq$ and $4p$ where $p,q$ are primes, as a special family of locally strictly simple quandles (i.e. quandles for which all proper subquandles are strictly simple). In particular we classify all latin quandles of size $pq$ and $4p$ and we show that latin quandles of size $8p$ are affine.

math.GR↗

arXiv2019

Principal and Doubly Homogeneous Quandles

In the paper we describe the class of principal quandles and show that connected quandles can be decomposed as a disjoint union of principal quandles. We also prove that simple affine quandles are finite and they can be characterized among finite simple quandles by several different equivalent properties, as for instance being doubly homogeneous (i.e. having a doubly transitive automorphism group). A complete description of finite doubly-homogeneous quandles is provided extending the result of \cite{V} and solving \cite[Problem 6.7]{2trans}. We also provide a classification of connected cyclic quandles with several fixed points independently from \cite{CQ}.

math.GR↗

arXiv2019

Involutive latin solutions of the Yang-Baxter equation

Wolfgang Rump showed that there is a one-to-one correspondence between nondegenerate involutive set-theoretic solutions of the Yang-Baxter equation and binary algebras in which all left translations $L_x$ are bijections, the squaring map is a bijection, and the identity $(xy)(xz) = (yx)(yz)$ holds. We call these algebras \emph{rumples} in analogy with quandles, another class of binary algebras giving solutions of the Yang-Baxter equation. We focus on latin rumples, that is, on rumples in which all right translations are bijections as well. We prove that an affine latin rumple of order $n$ exists if and only if $n=p_1^{p_1 k_1}\cdots p_m^{p_m k_m}$ for some distinct primes $p_i$ and positive integers $k_i$. A large class of affine solutions is obtained from nonsingular near-circulant matrices $A$, $B$ satisfying $[A,B]=A^2$. We characterize affine latin rumples as those latin rumples for which the displacement group generated by $L_x L_y\inv$ is abelian and normal in the group generated by all translations. We develop the extension theory of rumples sufficiently to obtain examples of latin rumples that are not affine, not even isotopic to a group. Finally, we investigate latin rumples in which the dual identity $(zx)(yx) = (zy)(xy)$ holds as well, and we show, among other results, that the generators $L_x L_y\inv$ of their displacement group have order dividing four.

math.GR↗

arXiv2019

On connected quandles of prime power order

We develop some general ideas to study connected quandles of prime power size and we classify non-affine connected quandles of size $p^3$ for $p>3$, using a combination of group theoretical and universal algebraic tools. As a byproduct we obtain a classification of Bruck loops of the same size and that they are in one-to-one correspondence with commutative automorphic loops.

math.GR↗

arXiv2018

On the Structure of Hom Quandles

We continue the study of the quandle of homomorphisms into a medial quandle begun in Crans and Nelson. We show that it suffices to consider only medial source quandles, and therefore the structure theorem of Jedlicka et al. provides a characterization of the Hom quandle. In the particular case when the target is 2-reductive this characterization takes on a simple form that makes it easy to count and determine the structure of the Hom quandle.

math.GT↗

arXiv2018

Simply connected latin quandles

A (left) quandle is connected if its left multiplication group acts transitively. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to so-called constant quandle cocycles that form a subset of quandle cocycles. A connected quandle is said to be \emph{simply connected} if it has no nontrivial coverings, or, equivalently, if its second constant cohomology groups are trivial. In this paper we develop a combinatorial approach to constant cohomology. Upon applying our theory, we prove that connected quandles that are affine over cyclic groups are simply connected (extending a result of Graña for quandles of prime size) and that finite doubly transitive quandles of order different from $4$ are simply connected.

math.GR↗

arXiv2017

Generalize Heisenberg Groups and Self-Duality

This paper compares two generalizations of Heisenberg groups and studies their connection to one of the major open problems in the field of locally compact abelian groups, namely the description of the self-dual locally compact abelian groups ([12]). The first generalization is presented by the so called generalized Heisenberg groups $\mathbb{H}(ω)$, defined in analogy with the classical Heisenberg group and the second one is inspired by the construction proposed by Mumford in [16] and named after him as Weyl-Mumford groups (WM groups). These two families can be defined also in the framework of topological groups. We investigate the relationship between locally compact WM groups, locally compact Generalized Heisenberg with center isomorphic to $\mathbb{T}$ and symplectic self-dualities.

math.GR↗