The star-triangle relation and the star-star relation as integrability conditions

Integrable models are significant examples for our understanding and they are elementary tools to get into physical reality. Their crucial roles are beyond physics and any investigation for the integrability of a model shares many interesting intersections with mathematics. One of the most famous examples of integrable models is the two-dimensional Ising model which is the simplification of ferromagnetic material. The question will be interested in in this thesis is what the star-triangle relation and the star-star relation are as integrability conditions for two-dimensional lattice spin models in statistical mechanics. The integrability conditions are particular equations that describe essential steps to be satisfied by the physical expressions such as Boltzmann weights in this case. Last decade, many non-trivial solutions to the star-triangle relation (STR) and the star-star relation (SSR) have been obtained and the subject is realized to be that it has many more connections with the fields of mathematical physics and even pure mathematics. The recently studied solutions are in terms of special functions such as lens elliptic hypergeometric gamma function, hyperbolic hypergeometric gamma function, and so on. In this thesis, there will be a discussion the explicit expressions of the solutions of lens hyperbolic hypergeometric gamma model. Also we get a journey in the relations of solutions to integrability conditions to other fields in the literature.