CyberSec Research

This paper addresses the interactions between three properties that a group algebra or more generally a pointed Hopf algebra may possess: being noetherian, having finite Gelfand-Kirillov dimension, and satisfying the Dixmier-Moeglin equivalence. First it is shown that the second and third of these properties are equivalent for group algebras $kG$ of polycyclic-by-finite groups, and are, in turn, equivalent to $G$ being nilpotent-by-finite. In characteristic 0, this enables us to extend this equivalence to certain cocommutative Hopf algebras. In the second and third parts of the paper finiteness conditions for group algebras are studied. In the second section we examine when a group algebra satisfies the Goldie conditions, while in the final section we discuss what can be said about a minimal counterexample to the conjecture that if $kG$ is noetherian then G is polycyclic-by-finite.

We consider algebraic identities for linear operators on associative algebras in which each term has degree 2 (the number of variables) and multiplicity 3 (the number of occurrences of the operator). We apply the methods of earlier work by the author and Elgendy which classified operator identities of degree 2, multiplicities 1 and 2. We begin with the general operator identity of multiplicity 3 which has 10 terms and indeterminate coefficients. We use the operadic concept of partial composition to generate all consequences of this identity in degree 3, multiplicity 4. The coefficient matrix of these consequences has size $105 \times 20$ and indeterminate entries. We compute the partial Smith form of this matrix and use Gr\"obner bases for determinantal ideals to discover which values of the indeterminates produce a matrix of submaximal rank. The only possible submaximal values of the rank are 16 and 19: there are 6 new identities of rank 16, and 8 new identities of rank 19.

In this article, we proceed on the transfer of the left endo-Noetherian property on certain ring extensions. We transfer of the right (left) endo-Noetherian property to the right (left) quotient rings. For a subring $T$ of $R$ and a finite set of indeterminates $X$, we prove that $T + XR[[X]]$ is left endo-Noetherian if and only if $R[[X]]$ is left endo-Noetherian. In addition, we prove that the subring $\Lambda :=\{ f \in R[[S,\omega ]]: f(1) \in T \}$ of the skew generalized power series ring $R[[S, \omega]]$ is left endo-Noetherian if and only if $R[[S, \omega]]$ is left endo-Noetherian. Also, we study the left endo-Noetherian property over the amalgamated duplication rings $R \bowtie I$ and $ R \bowtie ^f J$. Finally, we introduce additional results on left endo-Noetherian rings.

Skew braces play a central role in the theory of set-theoretic non-degenerate solutions of the Yang--Baxter equation, since their algebraic properties significantly affect the behaviour of the corresponding solutions (see for example [Ballester-Bolinches et al., Adv. Math. 455 (2024), 109880]). Recently, the study of nilpotency-like conditions for the solutions of the Yang--Baxter equation has drawn attention to skew braces of abelian type in which every substructure is an ideal (so-called, Dedekind skew braces); see for example [Ballester-Bolinches et al., Result Math. 80 (2025), Article Number 21]. The aim of this paper is not only to show that the hypothesis the skew brace is of abelian type can be neglected in essentially all the known results in this context, but also to extend this theory to skew braces whose additive or multiplicative groups are locally cyclic (and more in general of finite rank). Our main results -- which are in fact much more general than stated here -- are as follows: (1) Every finite Dedekind skew brace is centrally nilpotent. (2) Every hypermultipermutational Dedekind skew brace with torsion-free additive group is trivial. (3) Characterization of a skew brace whose additive or multiplicative group is locally cyclic (4) If a set-theoretic non-degenerate solution of the Yang--Baxter equation has a Dedekind structure skew brace and fixes the diagonal elements, then such a solution must be the twist solution.

We study maximal sublattices of finite semidistributive lattices via their complements. We focus on the conjecture that such complements are always intervals, which is known to be true for bounded lattices. Since the class of semidistributive lattices is the intersection of classes of join- and meet-semidistributive lattices, we study also complements for these classes, and in particular convex geometries of convex dimension 2, which is a subclass of join-semidistributive lattices. In the latter case, we describe the complements of maximal sublattices completely, as well as the procedure of finding all complements of maximal sublattices.

In this paper, we describe the non-commutative formal geometry underlying a certain class of discrete integrable systems. Our main example is a non-commutative analog, labeled $q$-P$(A_3)$, of the sixth $q$-Painlev\'{e} equation. The system $q$-P$(A_3)$ is constructed by postulating an extended birational representation of the extended affine Weyl group $\widetilde{W}$ of type $D_5^{(1)}$ and by selecting the same translation element in $\widetilde{W}$ as in the commutative case. Starting from this non-commutative discrete system, we develop a non-commutative version of Sakai$'$s surface theory, which allows us to derive the same birational representation that we initially postulated. Moreover, we recover the well-known cascade of multiplicative discrete Painlev\'e equations rooted in $q$-P$(A_3)$ and establish a connection between $q$-P$(A_3)$ and the non-commutative $d$-Painlev\'e systems introduced in I. Bobrova. Affine Weyl groups and non-Abelian discrete systems: an application to the $d$-Painlev\'e equations.

We introduce a new point of view to present classical notions related to set-theoretic solutions of the Yang-Baxter equation: left skew braces, dirings, left skew rings. The idea is, in the definition of a left near-ring $G$, to replace the multiplication of $G$ with two operations on the additive group $G$, one associative and the other left distributive. Two algebraic structures naturally appear: left skew rings and left weak rings, whose categories turn out to be canonically isomorphic.

We prove that coherent configurations can be represented as modules over Frobenius structures in the category of real nonnegative matrices. We generalize the notion of admissible morphism from association schemes to coherent configurations. We show that the Frobenius structure associated to a coherent configuration can be modified to become a dagger Frobenius structure, and use this to connect the coherent configurations to groupoids and $H^*$-algebras. We examine the properties of the dagger Frobenius structure with respect to admissible morphisms. We introduce the matrix $O$ obtained as the composition of comultiplication and multiplication of the dagger Frobenius structure and prove that we may obtain the valencies of colors, and thus recover the original coherent configuration, as an eigenvector of $O$. In the last part of the paper, we examine the spectrum of $O$ and apply it to generalize the Lagrange theorem from groups to association schemes.

I give a proof of Zel'manov's theorem that if $L$ is an $n$-Engel Lie algebra over a field $F$ of characteristic zero then $L$ is (globally) nilpotent. This is a very important result which extends Kostrikin's theorem that $L$ is locally nilpotent if the characteristic of $F$ is zero or some prime $p>n$. Zel'manov's proof contains some striking original ideas, and I wrote this note in an effort to understand his arguments. I hope that my efforts will be of use to other mathematicians in understanding this remarkable theorem.

We construct large classes of maximal commutative subalgebras in prime Steinberg algebras, generalizing a known result for Leavitt path algebras.

We investigate whether the group algebra of a finite group over a localisation of the integers is semiperfect. The main result is a necessary and sufficient arithmetic criterion in the ordinary case. In the modular case, we propose a conjecture, which extends the criterion.

Symmetry is a key property of numerical methods. The geometric properties of symmetric schemes make them an attractive option for integrating Hamiltonian systems, whilst their ability to exactly recover the initial condition without the need to store the entire solution trajectory makes them ideal for the efficient implementation of Neural ODEs. In this work, we present a Hopf algebraic approach to the study of symmetric B-series methods. We show that every B-series method can be written as the composition of a symmetric and "antisymmetric" component, and explore the structure of this decomposition for Runge-Kutta schemes. A major bottleneck of symmetric Runge-Kutta schemes is their implicit nature, which requires solving a nonlinear system at each step. By introducing a new set of order conditions which minimise the antisymmetric component of a scheme, we derive what we call Explicit and Effectively Symmetric (EES) schemes -- a new class of explicit Runge-Kutta schemes with near-symmetric properties. We present examples of second-order EES schemes and demonstrate that, despite their low order, these schemes readily outperform higher-order explicit schemes such as RK4 and RK5, and achieve results comparable to implicit symmetric schemes at a significantly lower computational cost.

In 1978, Frost and Storey asserted that a bivariate polynomial matrix is equivalent to its Smith normal form if and only if the reduced minors of all orders generate the unit ideal. In this paper, we first demonstrate by constructing an example that for any given positive integer s with s >= 2, there exists a square bivariate polynomial matrix M with the degree of det(M) in y equal to s, for which the condition that reduced minors of all orders generate the unit ideal is not a sufficient condition for M to be equivalent to its Smith normal form. Subsequently, we prove that for any square bivariate polynomial matrix M where the degree of det(M) in y is at most 1, Frost and Storey's assertion holds. Using the Quillen-Suslin theorem, we further extend our consideration of M to rank-deficient and non-square cases.

This is an introduction to the group algebras of the symmetric groups, written for a quarter-long graduate course. After recalling the definition of group algebras (and monoid algebras) in general, as well as basic properties of permutations, we introduce several families of elements in the symmetric group algebras $\mathbf{k}[S_n]$ such as the Young--Jucys--Murphy elements, the (sign-)integrals and the conjugacy class sums. Then comes a chapter on group actions and representations in general, followed by the core of this text: a study of the representations of symmetric groups (i.e., of left $\mathbf{k}[S_n]$-modules), including the classical theory of Young tableaux and Young symmetrizers. We prove in detail the main facts including the characterization of irreducible representations (in characteristic $0$), the Garnir relations, the standard basis theorem, the description of duals of Specht modules, and the hook length formula, as well as a number of less known results. Finally, we describe several bases of $\mathbf{k}[S_n]$ that arise from the study of Specht modules, including the Murphy cellular bases. The methods used are elementary and computational. We aim to assume as little as possible of the base ring $\mathbf{k}$, and to use as little as possible from representation theory (nothing more advanced than Maschke and Jordan--H\"older). Over 100 exercises (without solutions) are scattered through the text.

Given a field $\mathbb{F}$, we introduce a novel group $SD(\mathbb{F})$ of its self-maps: the solutions $f \colon \mathbb{F} \twoheadrightarrow \mathbb{F}$ to the functional equation $f \left( (x+y)/(x-y) \right) = (f(x) + f(y))/(f(x) - f(y))$ for all $ x \neq y$ in $\mathbb{F}$. We compute this group for all fields algebraic over $\mathbb{F}_p$. In particular, this group distinguishes $\mathbb{F}_5$ among all finite fields $\mathbb{F}_q$, and in fact among all subfields of $\overline{\mathbb{F}_q}$.

A curled algebra is a non-associative algebra in which $x$ and $x^2$ are linearly dependent for every element $x$. An algebra is called endo-commutative, if the square mapping from the algebra to itself preserves multiplication. In this paper, we provide a necessary and sufficient condition for a 3-dimensional curled algebra over an arbitrary field to be endo-commutative, expressed in terms of the properties of its underlying linear basis.

We study an analogue of the Andreadakis-Johnson filtration for automorphism groups of free algebras and introduce the notion of tangent Lie algebras for certain automorphism groups, defined as subalgebras of the Lie algebra of derivations. We show that, for many classical varieties of algebras, the tangent Lie algebra is contained in the Lie algebra of derivations with constant divergence. We also introduce the concepts of approximately tame and absolutely wild automorphisms of free algebras in arbitrary varieties and employ tangent Lie algebras to investigate their properties. It is shown that nearly all known examples of wild automorphisms of free algebras are absolutely wild -- with the notable exceptions of the Nagata and Anick automorphisms. We show that the Bergman automorphism of free matrix algebras of order two is absolutely wild. Furthermore, we prove that free algebras in any variety of polynilpotent Lie algebras -- except for the abelian and metabelian varieties -- also possess absolutely wild automorphisms.

Let $\mathfrak g$ be a classical simple Lie algebra over an algebraically closed field $\mathbb F$ of characteristic zero or large enough, and let $\mathfrak n$ be a maximal nilpotent subalgebra of $\mathfrak g$. The main tool in representation theory of $\mathfrak n$ is the orbit method, which classifies primitive ideals in the universal enveloping algebra ${\rm U}(\mathfrak n)$ and unitary representations of the unipotent group $N=\exp(\mathfrak n)$ in terms of coadjoint orbits on the dual space $\mathfrak n^*$. In the paper, we describe explicitly coadjoint orbits of low dimension for $\mathfrak n$ as above. The answer is given in terms of subsets of positive roots. As a corollary, we provide a way to calculate the number of irreducible complex representations of dimensions $q$, $q^2$ and $q^3$ for a maximal unipotent subgroup $N(q)$ in a classical Chevalley group $G(q)$ over a finite field $\mathbb F_q$ with $q$ elements. It turned out that this number is a polynomial in $q-1$ with nonnegative integer coefficients, which agrees with Isaac's conjecture.