It is known that the Temperley-Lieb algebra can be used to construct solutions of the Yang-Baxter equation, by means of a procedure known as Baxterization. The associated R-matrices, solutions of the Yang-Baxter equation, can be used to construct families of commuting transfer matrices. These transfer matrices, which contain integrable spin chains, can in principle be diagonalized by the algebraic Bethe ansatz technique (ABA). However, the R-matrices obtained from higher-spin representations of the Temperley-Lieb algebra lead to unusual commutation relations for the generators of the Yang-Baxter and reflection algebras, which has prevented the use of the ABA for such cases. Here we will present progress in the implementation of the ABA for open chains with free boundary conditions, which are characterized by simple reflection matrices, considering the spin-1 case in detail. That is, we will show how the reflection algebra can be used to construct off-shell Bethe states starting from a given reference state. We will also comment on scalar product formulas for the Bethe vectors.
Alexi Morin-Duchesne
