Solutions Of Yang-Baxter Equation Research Articles

Two theorems on balanced braces

AbstractTwo theorems of Gateva-Ivanova [Set-theoretic solutions of the Yang-Baxter equation, braces and symmetric groups, Adv. Math. 338 (2018), 649–701] on square-free set-theoretic solutions to the Yang–Baxter equation are extended to a wide class of solutions. The square-free hypothesis is almost completely removed. Gateva-Ivanova and Majid's ‘cyclic’ condition ${\boldsymbol {\rm lri}}$ is shown to be equivalent to balancedness, introduced in Rump [A decomposition theorem for square-free unitary solutions of the quantum Yang-Baxter equation, Adv. Math. 193 (2005), 40–55]. Basic results on balanced solutions are established. For example, it is proved that every finite, not necessarily square-free, balanced brace determines a multipermutation solution.

ℤ3 parafermionic chain emerging from Yang-Baxter equation.

We construct the 1D parafermionic model based on the solution of Yang-Baxter equation and express the model by three types of fermions. It is shown that the parafermionic chain possesses both triple degenerate ground states and non-trivial topological winding number. Hence, the parafermionic model is a direct generalization of 1D Kitaev model. Both the and model can be obtained from Yang-Baxter equation. On the other hand, to show the algebra of parafermionic tripling intuitively, we define a new 3-body Hamiltonian based on Yang-Baxter equation. Different from the Majorana doubling, the holds triple degeneracy at each of energy levels. The triple degeneracy is protected by two symmetry operators of the system, ω-parity P and emergent parafermionic operator Γ, which are the generalizations of parity PM and emergent Majorana operator in Lee-Wilczek model, respectively. Both the parafermionic model and can be viewed as SU(3) models in color space. In comparison with the Majorana models for SU(2), it turns out that the SU(3) models are truly the generalization of Majorana models resultant from Yang-Baxter equation.

On the structure and representations of non-balanced quantum doubles

For a field k and two finite groups G and X, when G acts on X from the right by group automorphisms, there is a Hopf algebra structure on k-space (kXop)* ⋇ kG (see Theorem 2.1), called a non-balanced quantum double and denoted by DX(G). In this paper, some Hopf algebra properties of DX(G) are given, the representation types of DX(G) viewed as a k-algebra are discussed, the algebra structure and module category over DX(G) are studied. Since the Hopf algebra structure of non-balanced quantum double DX(G) generalizes the usual quantum double D(G) for a finite group G, all results about DX(G) in this paper can also be used to describe D(G) as a special case and the universal R-matrix of DX(G) provides more solutions of Yang-Baxter equation.

The Right Braids, Quasi-braided Pre-tensor Categories, and General Yang–Baxter Operators#

This work is a development of braids, tensor categories and Yang–Baxter operators. According to Li [Li, F. (1998). Weak Hopf algebras and some new solution of quantum Yang-Baxter equation. J. Algebra 208:72–100; Li, F. (2000). Solutions of Yang-Baxter equation in endomorphism semigroup and quasi-(co)braided almost bialgebras. Comm. Algebra 28(5):2253–2270], it can be seen as a continuation of studying (not necessarily invertible) solutions of the (quantum) Yang–Baxter equation. We firstly introduce the right braid monoids and discuss their properties. Then, we define pre-tensor categories, pre-tensor functors and quasi-braided pre-tensor categories, and investiage their characterizations. Three examples are given from respectively a weak Hopf algebra, a crossed S-set of a Clifford monoid and the (strict) right braid category. Two universalities of the (strict) right braid category are gotten in order to characterize a category of general Yang–Baxter operators and a quasi-braided pre-tensor category. In a pre-tensor category we build a general centre of a pre-tensor category as a generalization of a centre and show that it is a quasi-braided pre-tensor category. At the end, a categorical interpretation of the quantum quasi-double of a weak Hapf algebra is obtained under a certain condition.

Solutions of yang-baxter equation in an endomorphism semigroup and quasi-(co)braided almost bialgebras*

The aim of this paper is to study the solutions of the Yang-Baxter equation in the endomorphism semigroup of the tensor product of a vector space. As preparation, we introduce the concepts of quasi-braided almost bialgebra (see also [10]) and quasi-cobraided almost bialgebra, and discuss some of their properties. In particular, it is shown that the quasi R-matrix R of every quasi-braided almost weak Hopf algebra is regular under von Neumann's meaning. The solutions of the Yang-Baxter equation in the endomorphism semigroups are constructed respectively from every quasi-braided almost bialgebra and every quasi-cobraided almost bialgebra. As examples, we explain how to build solutions of the Yang-Baxter equation from some weak Hopf algebras and all Clifford monoids. Finally, the FRT construction is given so as to build every solution of the Yang-Baxter equation from a quasi-cobraided bialgebra.

Another solution of Yang-Baxter equation on set and “metahomomorphisms on groups”

1 Another solution of Yang-Baxter equation on setIN ref [1], Drinfeld said that up to now there exist just two examples on the solution R of theYang-Baxter equation

Solutions of Yang-Baxter equation with color parameters

All solutions of Yang-Baxter equation with color parameters for six-vertex model are achieved by five solution transformations, three nondegenerate basic solutions and several degenerate basic solutions. It is shown that all solutions of Yang-Baxter equation with spectral parameter for six-vertex model are obtainable from solutions of Yang-Baxter equation with color parameters for six-vertex model.

From Braid Group Representations with Four Eigenvalues to Solutions of Yang-Baxter Equation11The project supported by National Natural Science Foundation of China and partially by Fork Ying Tong Education Foundation

A recursive formula of Jimbo-type trigonometric Yang-Baxterization is presented. The consistency conditions of the Yang-Baxterization for four eigenvalues are obtained. Several solutions of Yang-Baxter equations are obtained from a braid group representation.

Solutions of Yang-Baxter equations with colours: standard and nonstandard

Colour-extended solutions of YBE associated with G=Bn, Cn and Dn are derived and the coloured Birman-Wenzl algebra (CB-WA) is established. The authors point out that the CB-WA still satisfies the redundance conditions of Murakami (1989). For nonstandard solution, they take the nontrivial colour-solution of D2 as an example. They find that the colour-constraints induce a new solution with q4=1, the implication of this is not clear.

General solutions of Rj1/2(x) and quantum group structure

A general form of solutions of braid relations associated with Vj(X)V1/2 for Su(2) is calculated. A Yang-Baxterization prescription is presented to generate the corresponding solutions of Yang-Baxter equations. The authors show that this general form of solutions gives rise to a new construction of quantum group structure even though the q-commutation relations are kept standard.

Quantum Group Construction of Non-standard R-matrix for Yang-Baxter Equation

By taking the concept of weight conservation into account, new solutions of Yang-Baxter equation without spectral parameter are obtained through the non-standard R-matrix on the tensorial space V1/2 ⊗ Vj (j ≠ 1/2) in terms of the irreducible representation of the quantum universal enveloping algebra SLq (2). The non-standard R-matrix R1/2 1 is obtained in an explicit form for j = 1.

New R(x)-Matrix, L(x)-Operator and Quantum Algebra in Spin Model11The project supported in part by National Natural Science Foundation of China

New solution of Yang-Bax ter equation for spin-1 ⊗ spin- tensorial space is derived based on the direct calculation and extended Yang-Baxterization prescription. It contains the standard solution as a special case. By employing Faddeev-Reshetikhin-Takhtajan approach we give the correspondent quantum algebraic structure. The Yang-Baxterization of the L-operator is also made explicitly.

New solutions of Yang-Baxter equations: Birman-Wenzl algebra and quantum group structures

New braid group representations associated with the fundamental representations of Bn, Cn and Dn are derived. They are shown to satisfy Birman-Wenzl algebra. By Yang-Baxterization the authors obtain new solutions of Yang-Baxter equations, including the twisted extensions. Quantum group structures related to these new solutions are explicitly given.

NEW QUANTUM ALGEBRA ASSOCIATED WITH THE EXOTIC SOLUTIONS OF YANG-BAXTER EQUATIONS FOR An

Starting from the exotic solutions of Yang-Baxter equation (YBE) associated with the fundamental representations of An the new type of quantum algebra is established.

STRUCTURE CONSTANTS OF QUANTUM ALGEBRA AND YANG-BAXTER RELATION FOR THE Zn-SYMMETRIC STATISTICAL MODEL

The Zn-symmetric solution of Yang-Baxter equation in statistical mechanics is analyzed with care, the structure constants of the corresponding quantum algebra are obtained naturally, and a direct proof for this solution is presented.