Related papers: Yang-Baxter operators need quantum entanglement to…
A connection between the Yang-Baxter relation for maps and the multi-dimensional consistency property of integrable equations on quad-graphs is investigated. The approach is based on the symmetry analysis of the corresponding equations. It…
Involutive non-degenerate set theoretic solutions of the Yang-Baxter equation are considered, with a focus on finite solutions. A rich class of indecomposable and irretractable solutions is determined and necessary and sufficient conditions…
We introduce non-degenerate solutions of the Yang-Baxter equation in the setting of symmetric monoidal categories. Our theory includes non-degenerate set-theoretical solutions as basic examples. However, infinite families of non-degenerate…
We find a method to construct iteratively from a non-degenerate involutive set-theoretic solution of the Yang-Baxter equation an infinite family of very large non-degenerate involutive set-theoretic solutions. In case the initial solution…
We present particularly simple new solutions to the Yang--Baxter equation arising from two--dimensional cyclic representations of quantum $SU(2)$. They are readily interpreted as scattering matrices of relativistic objects, and the quantum…
New solutions of the quantum Yang-Baxter equation, depending in general on three arbitrary parameters, are written down. They are based on the root of unity representations of the quantum orthosymplectic superalgebra \\U, which were found…
Entanglement is a special feature of the quantum world that reflects the existence of subtle, often non-local, correlations between local degrees of freedom. In topological theories such non-local correlations can be given a very intuitive…
In our preceding papers we started considering the categories of tangles with flat G-connections in their complements, where G is a simple complex algebraic group. The braiding (or the commutativity constraint) in such categories satisfies…
Two types of Yang-Baxter systems play roles in the theoretical physics -- constant and colour dependent. The constant systems are used mainly for construction of special Hopf algebra while the colour or spectral dependent for construction…
The quantum Yang-Baxter equation admits generalisations to systems of Yang-Baxter type equations called Yang-Baxter systems. Starting from algebra structures, we propose new constructions of some constant as well as the spectral-parameter…
Yang-Baxter system related to quantum doubles is introduced and large class of both continuous and discrete symmetries of the solution manifold are determined. Strategy for solution of the system based on the symmetries is suggested and…
For any algebra two families of coloured Yang-Baxter operators are constructed, thus producing solutions to the two-parameter quantum Yang-Baxter equation. An open problem about a system of functional equations is stated. The matrix forms…
A new class of indecomposable, irretractable, involutive, non-degenerate set-theoretic solutions of the Yang--Baxter equation is constructed. This class complements the class of such solutions constructed in \cite{CO22} and together they…
The most common geometric interpretation of the Yang-Baxter equation is by braids, knots and relevant Reidemeister moves. So far, cubes were used for connections with the third Reidemeister move only. We will show that there are…
We construct $2^n$-families of solutions of the Yang-Baxter equation from $n$-products of three-dimensional $R$ and $L$ operators satisfying the tetrahedron equation. They are identified with the quantum $R$ matrices for the Hopf algebras…
Starting from a quantum dilogarithm over a Pontryagin self-dual LCA group $A$, we construct an operator solution of the Yang-Baxter equation generalizing the solution of the Faddeev-Volkov model. Based on a specific choice of a subgroup…
A biquandle is a solution to the set-theoretical Yang-Baxter equation, which yields invariants for virtual knots such as the coloring number and the state-sum invariant. A virtual biquandle enriches the structure of a biquandle by…
We analyze the connections between the mathematical theory of knots and quantum physics by addressing a number of algorithmic questions related to both knots and braid groups. Knots can be distinguished by means of `knot invariants', among…
While there is a general consensus about the structure of one qubit operations in topological quantum computer, two qubits are as usual a more difficult and complex story of different attempts with varying approaches, problems and…
Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set $X$ and a function r:X x X --> X x X which satisfies the braid relation. We…