Related papers: Solutions of the Quantum-Yang-Baxter-Equation from…
A three-parametric $R$-matrix satisfying a graded Yang-Baxter equation is introduced.This $R$-matrix allows us to construct new quantum supergroups which are deformations of the supergroup $GL(1/1)$ and the universal enveloping algebra…
In this paper we study the quantum Clifford-Hopf algebras $\widehat{CH_q(D)}$ for even dimensions $D$ and obtain their intertwiner $R-$matrices, which are elliptic solutions to the Yang- Baxter equation. In the trigonometric limit of these…
We present solutions for the (constant and spectral-parameter) Yang-Baxter equations and Yang-Baxter systems arising from algebra structures and discuss about their symmetries. In the last section, we present some applications.
Let $r:X^{2}\rightarrow X^{2}$ be a set-theoretic solution of the Yang-Baxter equation on a finite set $X$. It was proven by Gateva-Ivanova and Van den Bergh that if $r$ is non-degenerate and involutive then the algebra $K\langle x \in X…
We describe a geometric construction of all nondegenerate trigonometric solutions of the associative and classical Yang-Baxter equations. In the associative case the solutions come from symmetric spherical orders over the irreducible nodal…
We introduce a new family of classical $r$-matrices for the Lie algebra $\mathfrak{sl}_n$ that lies in the Zariski boundary of the Belavin-Drinfeld space ${\mathcal M}$ of quasi-triangular solutions to the classical Yang-Baxter equation. In…
We present resent results regarding invertible, non-degenerate solutions of the set-theoretic Yang-Baxter and reflection equations. We recall the notion of braces and we present and prove various fundamental properties required for the…
We quantize the interaction of gravity with Yang-Mills and spinor fields, hence offering a quantum theory incorporating all four fundamental forces of nature. Using canonical quantization we obtain solutions of the Wheeler-DeWitt equation…
Bethe Ansatz was discoverd in 1932. Half a century later its algebraic structure was unearthed: Yang-Baxter equation was discovered, as well as its multidimensional generalizations [tetrahedron equation and $d$-simplex equations]. Here we…
We consider general relativity with cosmological constant minimally coupled to electromagnetic field and assume that four-dimensional space-time manifold is the warped product of two surfaces with Lorentzian and Euclidean signature metrics.…
We have constructed series of the spectral parameter dependent solutions to the Yang-Baxter equations defined on the tensor product of reducible representations with symmetry of quantum algebra. These series are produced as descendant…
We investigate non-commutative differential calculus on the supersymmetric version of quantum space where the non-commuting super-coordinates consist of bosonic as well as fermionic (Grassmann) coordinates. Multi-parametric quantum…
We define a new class of unitary solutions to the classical Yang-Baxter equation (CYBE). These ``boundary solutions'' are those which lie in the closure of the space of unitary solutions to the modified classical Yang-Baxter equation…
We present a formula for trigonometric orthosymplectic $R$-matrices associated with any parity sequence, and establish their factorization into the ordered product of $q$-exponents parametrized by positive roots in the corresponding reduced…
Motivated by the study of the operator forms of the constant classical Yang-Baxter equation given by Semonov-Tian-Shansky, Kupershmidt and the others, we try to construct the rational solutions of the classical Yang-Baxter equation with…
The theory of the set-theoretic Yang-Baxter equation is reviewed from a purely algebraic point of view. We recall certain algebraic structures called shelves, racks and quandles. These objects satisfy a self-distributivity condition and…
The new method of solving quantum mechanical problems is proposed. The finite, i.e. cut off, Hilbert space is algebraically implemented in the computer code with states represented by lists of variable length. Complete numerical solution of…
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…
A variety of Yang-Baxter maps are obtained from integrable multi-field equations on quad-graphs. A systematic framework for investigating this connection relies on the symmetry groups of the equations. The method is applied to lattice…
Explicit solutions of the quantum Yang-Baxter equation are given corresponding to the non-unitary solutions of the classical Yang-Baxter equation for sl(5).