Combinatorics of Lax Objects in Bethe Ansatz

Algebraic Bethe Ansatz, also known as quantum inverse scattering method, is a consistent tool based on the Yang-Baxter equation which allows to construct Bethe Ansatz exact solutions. One of the most important objects in algebraic Bethe Ansatz is a monodromy matrix M̂, which is defined as an appropriate product of so-called Lax operators L̂ (local transition operators). Monodromy matrix as well as each of Lax operators acts in the tensor product of the quantum space with an auxiliary space ℂ². Thus M̂, when written in the standard basis of auxiliary space, consists of four elements Â, B̂, Ĉ, D̂, which are the operators acting in quantum space , where B̂ and Ĉ are step operators and the remaining generate all constants of motion. In this work a consistent method of construction of the Bethe Ansatz eigenstates in terms of objects â, b̂, ĉ, d̂ i.e. matrix elements of the Lax operators in the auxiliary space is proposed.