We discuss the higher-dimensional solutions of the Yang-Baxter equation and reflection equation in the elliptic case. The Hamiltonian of a spin-1 quantum anisotropic chain with non-trivial boundary term is given.
Some special solutions to the reflection equation are considered. These boundary matrices are defined on the common quantum space with the other operators in the chain. The relations with the Drinfeld twist are discussed. PACS: 02.90.+p, 03.20.+i,03.65.Fd, 11.30.-j 0 1 On a covariance property of the reflection equation Let the operator-valued function R(u) : C → Aut(V ⊗ V) be a solution to the quantum Yang-Baxter equation [1] in a finite-dimensional linear space V. Let us connect with a given matrix R(u) two associative algebras TR and UR generated by noncommutative entries of the square matrices T(u) and K(u) satisfying the fundamental commutator relation [1, 2] or the reflection equation [3, 4] R12(u − v) 1
All regular 4×4 solutions of the Yang-Baxter equation
The author considers solutions of the Yang-Baxter equation such that the logarithmic derivative of the transfer matrix yields a quantum spin Hamiltonian which is isotropic in spin space, i.e. SU(2)-invariant. Four such solutions are known for each value of the spin S. (For S=1/2 they degenerate into the same solution, and for S=1 they only give three different solutions). For S<or=6 he shows that these are the only solutions which are SU(2)-invariant, except for S=3 when there is a fifth solution.