Using results of Ellis-Rodríguez Fernández, an explicit description by generators and relations is given of the mod q Schur multiplier, and this is shown to be the kernel of a universal q-central extension in the case of a q-perfect group, i.e. one which is generated by commutators and q-th powers. These results generalise earlier work [by] K. Dennis and Brown-Loday.

This paper is devoted to new algebraic structures, called qualgebras and squandles. Topologically, they emerge as an algebraic counterpart of knotted 3-valent graphs, just like quandles can be seen as an "algebraization" of knots. Algebraically, they are modeled after groups with conjugation and multiplication/squaring operations. We discuss basic properties of these structures, and introduce and study the notions of qualgebra/squandle 2-cocycles and 2-coboundaries. Knotted 3-valent graph invariants...

Soit $c\>1$, $p$ un nombre premier et $𝒜$ une partie de $\mathbb{Z}/p\mathbb{Z}$ de cardinal supérieur à $c\sqrt{p}$ telle que pour tout sous-ensemble non vide $ℬ$ de $𝒜$, on a ${\sum }_{b\in ℬ}b\ne 0$. On montre qu’il existe $s$ premier à $p$ tel que l’ensemble $s.𝒜$ est très concentré autour de l’origine et qu’il est presque entièrement composé d’éléments de partie fractionnaire positive. Plus précisément, on a$$\sum _{a\in 𝒜}∥\frac{sa}{p}∥\<1+O(p^{-1/4}lnp)\text{et}\sum _{\begin{array}{c}a\in 𝒜,\\ \{sa/p\}\ge 1/2\end{array}}∥\frac{sa}{p}∥=O(p^{-1/4}lnp).$$On montre également que les termes d’erreurs ne peuvent être remplacés par $o\left(p^{-1/2}\right)$.

This article establishes the algebraic covering theory of quandles. For every connected quandle Q with base point q ∈ Q, we explicitly construct a universal covering p: (Q̃,q̃̃) → (Q,q). This in turn leads us to define the algebraic fundamental group $\pi ₁(Q,q):=Aut\left(p\right)=g\in Adj{\left(Q\right)}^{\text{'}}|q^g=q$, where Adj(Q) is the adjoint group of Q. We then establish the Galois correspondence between connected coverings of (Q,q) and subgroups of π₁(Q,q). Quandle coverings are thus formally analogous to coverings of topological spaces, and resemble Kervaire’s...

We describe how the constructions of quantum homogeneous spaces using infinitesimal invariance and quantum coisotropic subgroups are related. As an example we recover the quantum 4-sphere of [2] through infinitesimal invariance with respect to ${}_q\left(SU\left(2\right)\right)$.

Quantum quasigroups and loops are self-dual objects that provide a general framework for the nonassociative extension of quantum group techniques. They also have one-sided analogues, which are not self-dual. In this paper, natural quantum versions of idempotence and distributivity are specified for these and related structures. Quantum distributive structures furnish solutions to the quantum Yang-Baxter equation.

We review the formulation and proof of the Baum-Connes conjecture for the dual of the quantum group $S{U}_q\left(2\right)$ of Woronowicz. As an illustration of this result we determine the K-groups of quantum automorphism groups of simple matrix algebras.