Related papers: Moufang sets and structurable division algebras

We establish a link between abelian regular subgroup of the affine group, and commutative, associative algebra structures on the underlying vector space that are (Jacobson) radical rings. As an application, we show that if the underlying…

We consider algebras with basis numerated by elements of a group $G.$ We fix a function $f$ from $G\times G$ to a ground field and give a multiplication of the algebra which depends on $f$. We study the basic properties of such algebras. In…

For the class of solvable groups of homeomorphisms of the line preserving orientation and containing a freely acting element, we establish the metabelianity of the quotient group $G/H_G$, where the elements of the normal subgroup $H_G$ are…

We show that a differential algebraic group can be filtered by a finite subnormal series of differential algebraic groups such that successive quotients are almost simple, that is have no normal subgroups of the same type. We give a…

Let $M$ be a regular matroid. The Jacobian group ${\rm Jac}(M)$ of $M$ is a finite abelian group whose cardinality is equal to the number of bases of $M$. This group generalizes the definition of the Jacobian group (also known as the…

We construct explicitly a class of coboundary Poisson-Lie structures on the group of formal diffeomorphisms of ${\Bbb R}^n$. Equivalently, these give rise to a class of coboundary triangular Lie bialgebra structures on the Lie algebra $W_n$…

By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group has a derangement (an element with no fixed points). The existence of derangements with additional properties has attracted much attention,…

Let a Moufang loop Q contain a non-unitary subloop, which is a simple loop. Then Q is not embedded into a loop of invertible elements of any alternative algebra.

Let $F$ be a field of characteristic not 2 or 3. The first Tits construction is a well-known tripling process to construct separable cubic Jordan algebras, especially Albert algebras. We generalize the first Tits construction by choosing…

It is well-known that every algebraic group of type F_4 is the automorphism group of an exceptional Jordan algebra, and that up to isogeny all groups of type ^1E_6 with trivial Tits algebras arise as the isometry groups of norm forms of…

We generalize the hierarchy construction to generic 2+1D topological orders (which can be non-Abelian) by condensing Abelian anyons in one topological order to construct a new one. We show that such construction is reversible and leads to a…

We study second-order modular differential equations whose solutions transform equivariantly under the modular group. In the reducible case, we construct all such solutions using an explicit ansatz involving Eisenstein series and the…

A subset $A$ of an additive abelian group is an $h$-Sidon set if every element in the $h$-fold sumset $hA$ has a unique representation as the sum of $h$ not necessarily distinct elements of $A$. Let $\mathbf{F}$ be a field of characteristic…

The Gruenberg-Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices $p$, $q$ are joined by an edge whenever the group has an element of…

Various descending chains of subgroups of a finite permutation group can be used to define a sequence of `basic' permutation groups that are analogues of composition factors for abstract finite groups. Primitive groups have been the…

Combinatorial objects such as rooted trees that carry a recursive structure have found important applications recently in both mathematics and physics. We put such structures in an algebraic framework of operated semigroups. This framework…

This is the second introductory paper concerning structures called rootoids and protorootoids, the definition of which is abstracted from formal properties of Coxeter groups with their root systems and weak orders. The ubiquity of…

The aim of this article is to discuss and clarify the notion of fractality for subgroups of the group of automorphisms of a regular rooted tree. For this purpose we define three types of fractality. We show that they are not equivalent, by…

It is shown that every separable abelian topological group is isomorphic with a topological subgroup of a monothetic group (that is, a topological group with a single topological generator). In particular, every separable metrizable abelian…

This paper contains a quite detailed description of the C*-algebra arising from the transformation groupoid of a rational map of degree at least two on the Riemann sphere. The algebra is decomposed stepwise via extensions of familiar…