Related papers: Free commutative skew braces
We give an alternative description of the top algebra of the free crossed square of algebras on 2-construction data in terms of tensors and coproducts of crossed modules of commutative algebras.
We prove a structure theorem for finite perfect two-sided skew braces. The main tool is a central product theory for skew braces, developed here in both external and internal form; we show that these two constructions are equivalent. Our…
The main objective of this paper is to study factorisations of skew left braces through abelian subbraces. We prove a skew brace theoretical analog of the classical It\^o's theorem about product of two abelian groups: if $B = A_1A_2$ is a…
This work investigates the combinatorial structures underlying cyclic conditional freeness and introduces cumulants that serve to linearize the cyclic conditional additive convolution. In the process, we establish the notion of "cyclic…
The concepts of evaluation and interpolation are extended from univariate skew polynomials to multivariate skew polynomials, with coefficients over division rings. Iterated skew polynomial rings are in general not suitable for this purpose.…
Our primary focus is on the theory of skew braces, specifically exploring their connection with combinatorial solutions to the Yang-Baxter equation. Skew braces have recently emerged as intriguing algebraic structures, and their link to the…
We introduce a notion of "freely braided element" for simply laced Coxeter groups. We show that an arbitrary group element $w$ has at most $2^{N(w)}$ commutation classes of reduced expressions, where $N(w)$ is a certain statistic defined in…
The aim of this paper is to study the group of isomorphism classes of torsors of finite flat group schemes of rank 2 over a commutative ring $R$. This, in particular, generalises the group of quadratic algebras (free or projective), which…
For a free partial action of a group in a set we realize the associated partial skew group ring as an algebra of functions with finite support over an equivalence relation and we use this result to characterize the ideals in the partial…
According to Letourmy and Vendramin, a representation of a skew brace is a pair of representations on the same vector space, one for the additive group and the other for the multiplicative group, that satisfies a certain compatibility…
We show that the centralizer of a nonscalar element in the coproduct $k\langle X\rangle *k[Y]$ of a free associative algebra and a polynomial algebra over a given field is commutative. For $k\langle X \rangle$ this is part of Bergman's…
We investigate structural and rigidity properties of \emph{Lie skew braces} (LSBs), objects essentially known in the literature as \emph{post--Lie groups}, obtained by endowing a manifold with two compatible group laws that share the same…
In this paper, we explore linear representations of skew left braces, which are known to provide bijective non-degenerate set-theoretical solutions to the Yang--Baxter equation that are not necessarily involutive. A skew left brace $(A,…
A skew brace $A = (A,\cdot,\circ)$ is said to be \textit{left-simple} if $A\neq1$ and it has no left ideal other than $1$ and $A$. The purpose of this paper is to give a partial classification of the finite left-simple skew braces. A result…
In this paper I consider the structure of the polylinear mapping of the free algebra over the commutative ring.
We continue the study of freely braided elements of simply laced Coxeter groups, which we introduced in a previous work (math.CO/0301104). A known upper bound for the number of commutation classes of reduced expressions for an element of a…
In this paper we generalize the well-known construction of shuffle product algebras by using mixable shuffles, and prove that any free Baxter algebra is isomorphic to a mixable shuffle product algebra. This gives an explicit construction of…
The classical theory of free analysis generalizes the noncommutative (nc) polynomials and rational functions, easily providing such results as an nc analogue of the Jacobian conjecture. However, the classical theory misses out on important…
In this paper we provide necessary and sufficient conditions for strongly group graded rings to be simple. For a strongly group graded ring $R = \bigoplus_{g\in G} R_g$ the grading group $G$ acts, in a natural way, as automorphisms of the…
Let X be a smooth projective surface defined over an uncountable algebraically closed field k and let k(X) be its field of rational functions. Let s be an automorphism of X. This paper proves there is a non-negative integer n and elements a…