We will present solutions to the constant Yang-Baxter equation, in any dimension $n$ . More precisely, for any $n$ , we will create an infinite family of $n^2$ by $n^2$ matrices which are solutions to the constant Yang-Baxter equation. The total number of non-vanishing entries of such a matrix is $4n^2$ for $n$ even, and $4(n-1)(n)+1$ for $n$ odd. We will also present the unitary conditions for those matrices. Moreover, we discuss the entangling property of those matrices.We will present solutions to the constant Yang-Baxter equation, in any dimension $n$. More precisely, for any $n$, we will create an infinite family of $n^2$ by $n^2$ matrices which are solutions to the constant Yang-Baxter equation. The total number of non-vanishing entries of such a matrix is $4n^2$ for $n$ even, and $4(n-1)(n)+1$ for $n$ odd. We will also present the unitary conditions for those matrices. Moreover, we discuss the entangling property of those matrices.

arXiv:1806.08400Jun 20, 20182 citations

Solving the two-dimensional constant quantum Yang–Baxter equation

Jarmo Hietarinta

A detailed analysis of the constant quantum Yang–Baxter equation Rk1k2j1j2 Rl1k3k1j3Rl2l3k2k3= Rk2k3j2j3 Rk1l3j1k3Rl1l2k1k2 in two dimensions is presented, leading to an exhaustive list of its solutions. The set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the Gröbner-basis methods. Each solution was then transformed into a canonical form (based on the various trace matrices of R) for final elimination of duplicates and subcases. If we use homogeneous parametrization the solutions can be combined into 23 distinct cases, modulo the well-known C, P, and T reflections, and rotations and scalings R̃=κ(Q⊗Q)R(Q⊗Q)−1.A detailed analysis of the constant quantum Yang–Baxter equation Rk1k2j1j2 Rl1k3k1j3Rl2l3k2k3= Rk2k3j2j3 Rk1l3j1k3Rl1l2k1k2 in two dimensions is presented, leading to an exhaustive list of its solutions. The set of 64 equations for 16 unknowns was first reduced by hand to several subcases which were then solved by computer using the Gröbner-basis methods. Each solution was then transformed into a canonical form (based on the various trace matrices of R) for final elimination of duplicates and subcases. If we use homogeneous parametrization the solutions can be combined into 23 distinct cases, modulo the well-known C, P, and T reflections, and rotations and scalings R̃=κ(Q⊗Q)R(Q⊗Q)−1.

Journal of Mathematical PhysicsMay 1, 199381 citations

New Elliptic Solutions of the Yang–Baxter Equation

Dmitry Chicherin, Sergey E. Derkachov, Vyacheslav Pavlovich Spiridonov

arXiv:1412.3383Communications in Mathematical PhysicsFeb 22, 201614 citations

We will present solutions to the constant Yang-Baxter equation, in any dimension

n

. More precisely, for any

n

, we will create an infinite family of

n^2

n^2

matrices which are solutions to the constant Yang-Baxter equation. The total number of non-vanishing entries of such a matrix is

4n^2

for

n

even, and

4(n-1)(n)+1

for

n

odd. We will also present the unitary conditions for those matrices. Moreover, we discuss the entangling property of those matrices.