Related papers: Bijective solutions to the Pentagon Equation
The Yang-Baxter equation (YBE) and the reflection equation (RE) both come from mathematical physics, and they can be defined in any monoidal category. For cartesian monoidal categories, we prove that every solution to the RE provides a…
Idempotent left nondegenerate solutions of the Yang-Baxter equation are in one-to-one correspondence with twisted Ward left quasigroups, which are left quasigroups satisfying the identity $(x*y)*(x*z)=(y*y)*(y*z)$. Using combinatorial…
Skew braces are algebraic structures related to the solutions of the set-theoretic quantum Yang-Baxter equation. We develop the central nilpotency theory for such algebraic structures in the sense of Freese-McKenzie \cite{comm} and we…
For Ore semigroups $P$ with an order unit, we prove that there is a bijection between $E_0$-semigroups over $P$ and product systems of $C^{*}$-correspondences over $P^{op}$. We exploit this bijection and show that the reduced…
As a generalization of skew braces, the notion of skew trusses was introduced by T. Brzezinski. It was shown that every Rota-Baxter group has the structure of skew braces by V. G. Bardakov and V. Gubarev. To investigate an analogue of…
We formulate the pentagon relation for quantum dilogarithm elements in the structure group of a quantum cluster scattering diagram (QCSD). As an application, we show the nonpositivity of a certain class of nonskew-symmetric QCSDs. Also, we…
We generalise the notions of semi-regularity, regularity, and bi-regularity to unitary solutions of the braided Pentagon equation in concrete W*-categories with semi-regular/regular/bi-regular braiding, and study their properties. We show,…
We find new solutions to the Yang--Baxter equation in terms of the intertwiner matrix for semi-cyclic representations of the quantum group $U_q(s\ell(2))$ with $q= e^{2\pi i/N}$. These intertwiners serve to define the Boltzmann weights of a…
This paper aims to determine the images of the braid group under representations afforded by the Yang Baxter equation when the solution is a nontrivial $4 \times 4$ matrix. Making the assumption that all the eigenvalues of the Yang Baxter…
In this article, we discuss the notion of partition of elements in an arbitrary Coxeter system $(W,S)$: a partition of an element $w$ is a subset $\mathcal P\subseteq W$ such that the left inversion set of $w$ is the disjoint union of the…
The orbit polytope for a finite group G acting linearly and freely on a sphere S is used to construct a cellularized fundamental domain for the action. A resolution of the integers over G results from the associated G-equivariant…
The semidirect product $\mathbb{G}=\mathbb{L}\rtimes \mathbb{K}$ attached to a compact-group action on a connected, simply-connected solvable Lie group has a dense set of compact elements precisely when the $s\in \mathbb{K}$ operating on…
First of all, we recall the well known notion of semidirect product both for classical algebraic structures (like groups and rings) and for more recent ones (digroups, left skew braces, heaps, trusses). Then we analyse the concept of…
We study noninvolutive set-theoretic solutions $(X,r)$ of the Yang-Baxter equations in terms of the properties of the canonically associated algebraic objects-the braided monoid $S(X,r)$, the quadratic Yang-Baxter algebra $A= A(\textbf{k},…
Let $S$ be a right group. Then there exist two congruences $\sim$ and $\equiv$ on $S$ such that $S$ is the product of its quotient semigroups $S/{\sim}$ and $S/{\equiv}$, where $S/{\sim}$ is a group and $S/{\equiv}$ is a right zero…
Building on the previous extensive study of Yang, Gould and the present author, we provide a more precise insight into the group-theoretical ramifications of the word problem for free idempotent generated semigroups over finite biordered…
The quantum Yang-Baxter equation is a braiding condition on vector spaces which is of high relevance in several fields of mathematics, such as knot theory and quantum group theory. Their combinatorial counterpart are set-theoretic solutions…
We study some relations between left cancellative left semi-braces and other existing algebraic structures. In particular, we show that every left semi-brace arises from a left seminear-ring, extending the correspondence given by Rump…
We prove that every countable group with solvable power problem embeds into a finitely presented 2-generated group with solvable power and conjugacy problems.
The analytic continuation of the Lippmann-Schwinger bras and kets is obtained and characterized. It is shown that the natural mathematical setting for the analytic continuation of the solutions of the Lippmann-Schwinger equation is the…