Related papers: Yang-Baxter operators in symmetric categories
Baxter operators are constructed for quantum spin chains with deformed $s\ell_2$ symmetry. The parallel treatment of Yang-Baxter operators for the cases of undeformed, trigonometrically and elliptically deformed symmetries presented earlier…
We develop a theory of non-unitary set-theoretical solutions to the Quantum Yang-Baxter equation. Our results generalize those obtained by Etingof, Schedler and the author. We remark that some of our constructions are similar to…
In the first part we recall two famous sources of solutions to the Yang-Baxter equation -- R-matrices and Yetter-Drinfel$'$d (=YD) modules -- and an interpretation of the former as a particular case of the latter. We show that this result…
The study of non-degenerate set-theoretic solutions of the Yang-Baxter equation calls for a deep understanding of the algebraic structure of a skew left brace. In this paper, the skew brace theoretical property of solubility is introduced…
We construct two $Osp(n|2m)$ solutions of the graded Yang-Baxter equation by using the algebraic braid-monoid approach. The factorizable S-matrix interpretation of these solutions is also discussed.
A new method to construct involutive non-degenerate set-theoretic solutions $(X^n,r^{(n)})$ of the Yang-Baxter equation from an initial solution $(X,r)$ is given. Furthermore, the permutation group $\mathcal{G}(X^n,r^{(n)})$ associated to…
The representation theory of the Drinfeld doubles of dihedral groups is used to solve the Yang-Baxter equation. Use of the 2-dimensional representations recovers the six-vertex model solution. Solutions in arbitrary dimensions, which are…
We study the deformations of a wide class of Yang-Baxter (YB) operators arising from Lie algebras. We relate the higher order deformations of YB operators to Lie algebra deformations. We show that the obstruction to integrating deformations…
Recently V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions of the quantum Yang-Baxter equation, i.e. solutions given by a permutation $R$ of the set…
The structure groups of non-degenerate symmetric set-theoretical solutions of the quantum Yang-Baxter equation provide an infinite family of Garside groups with many interesting properties. Given a non-degenerate symmetric solution, we…
Braces and linear cycle sets are algebraic structures playing a major role in the classification of involutive set-theoretic solutions to the Yang-Baxter equation. This paper introduces two versions of their (co)homology theories. These…
We develop a method to construct all the indecomposable involutive set-theoretic solutions of the Yang-Baxter equation with a prime-power number of elements and cyclic permutation group. Moreover, we give a complete classification of the…
A new type of algebras that represent a generalization of both quantum groups and braided groups is defined. These algebras are given by a pair of solutions of the Yang--Baxter equation that satisfy some additional conditions. Several…
By means of contractions of Lie algebras, we obtain new classes of indecomposable quasi-classical Lie algebras that satisfy the Yang-Baxter equations in its reformulation in terms of triple products. These algebras are shown to arise…
We introduce the concept of an extended O-operator that generalizes the well-known concept of a Rota-Baxter operator. We study the associative products coming from these operators and establish the relationship between extended O-operators…
The unitary braiding operators describing topological entanglements can be viewed as universal quantum gates for quantum computation. With the help of the Brylinskis's theorem, the unitary solutions of the quantum Yang--Baxter equation can…
We develop a categorical approach to the dynamical Yang-Baxter equation (DYBE) for arbitrary Hopf algebras. In particular, we introduce the notion of a dynamical extension of a monoidal category, which provides a natural environment for…
We complete the classification of $4\times 4$ regular solutions of the Yang-Baxter equation. Apart from previously known models, we find four new models of non-difference form. All the new models give rise to Hamiltonians and transfer…
We study certain non-symmetric wavefunctions associated to the quantum nonlinear Schr\"odinger model, introduced by Komori and Hikami using Gutkin's propagation operator, which involves representations of the degenerate affine Hecke…
We prove that the category of solutions of the set-theoretic Yang-Baxter equation of Frobenius-Separability (FS) type is equivalent to the category of pointed Kimura semigroups. As applications, all involutive, idempotent, nondegenerate,…