Related papers: Free commutative skew braces
A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator…
In this paper, we observe the amalgamated free product structure of a Graph W*-probability space. In [16] and [17], we already observed the operator-valued freeness conditions on a graph W*-algebra. By using the conditions, we will consider…
In this article we describe the $G\times G$-equivariant $K$-ring of $X$, where $X$ is a regular compactification of a connected complex reductive algebraic group $G$. Furthermore, in the case when $G$ is a semisimple group of adjoint type,…
Given a group $G$ and a subset $X \subset G$, an element $g \in G$ is called quasi-positive if it is equal to a product of conjugates of elements in the semigroup generated by $X$. This notion is important in the context of braid groups,…
We show that stable commutator length is rational on free products of free Abelian groups amalgamated over $\mathbb{Z}^k$, a class of groups containing the fundamental groups of all torus knot complements. We consider a geometric model for…
We study the ring theoretical structures of mixable shuffle algebras and their associated free commutative Rota-Baxter algebras. For this study we utilize the connection of the mixable shuffle algebras with the overlapping shuffle algebra…
In this paper we construct free Hom-semigroups when its unary operation is multiplicative and is an involution. Our method of construction is by bracketed words. As a consequence, we obtain free Hom-associative algebras generated by a set…
Skew braces are one of the main algebraic tools controlling the structure of a non-degenerate bijective set-theoretic solution of the Yang-Baxter equation. The aim of this paper is to study model-theoretically tame skew braces, with…
Dynamical skew braces are known to produce solutions to the quiver-theoretic Yang--Baxter equation. Under a technical hypothesis, we prove that these solutions are braided groupoids (and hence skew bracoids in the sense of Sheng, Tang and…
Say that a cone is a commutative monoid in which x+y=0 implies that x=y=0. We show that cones (resp. simple cones) of many kinds order-embed or even embed unitarily into refinement cones (resp. simple refinement cones) of the same kind,…
The purpose of this paper is to make the theory of vertex algebras trivial. We do this by setting up some categorical machinery so that vertex algebras are just ``singular commutative rings'' in a certain category. This makes it easy to…
We define combinatorial representations of finite skew braces and use this idea to produce a database of skew braces of small size. This database is then used to explore different concepts of the theory of skew braces such as ideals, series…
Let $R$ be an algebra over a commutative ring $k$. Suppose that $R$ is endowed with a descending filtration indexed on an ordered group $(G,<)$ such that the restriction to $k$ is positive. We show that the existence of free algebras on a…
The aim of this article is to advance the knowledge on the theory of skew left braces. We introduce a subclass of skew left braces, which we denote by $\mathcal{I}_n$, $n \ge 1$, such that elements of the annihilator and lower central…
We introduce an embedding of the free magma on a set A into the direct product of the free magma on a singleton set and the free semigroup on A. This embedding is then used to prove several theorems related to algebraic independence of…
Suppose $R$ is a commutative ring with identity and a fixed invertible element $q^{\frac{1}{2}}$ such that $q+q^{-1}$ is invertible. For an oriented surface $\Sigma$, let $\mathcal{S}(\Sigma;R)$ denote the Kauffman bracket skein algebra of…
For an arbitrary rational polyhedron we consider its decompositions into Minkowski summands and, dual to this, the free extensions of the associated pair of semigroups. Being free for the pair of semigroups is equivalent to flatness for the…
We determine all the multiplicity-free representations of the symmetric group. This project is motivated by a combinatorial problem involving systems of set-partitions with a specific pattern of intersection.
In this paper, we consider the relation between the group freeness and the amalgamated freeness of crossed product algebras.
In earlier work, the author classified rigid representations of a quiver by finitely generated free modules over a principal ideal ring. Here we extend the results to representations of a quiver by finitely generated projective modules over…