Unifying Approaches in Integrable Systems: Quantum and Statistical, Ultralocal and Nonultralocal

The aim of this review is to present the list of by now a significant collection of quantum integrable models, ultralocal as well as nonultralocal, in a systematic way stressing on their underlying unifying algebraic structures. We restrict to quantum and statistical models belonging to trigonometric and rational classes with (2 x 2)- Lax operators. The ultralocal models can be classified successfully through their associated quantum algebra and are governed by the Yang-Baxter equation, while the nonultralocal models, the theory of which is still in the developmental stage, allow systematization through the braided Yang-Baxter equation. Along with the known integrable models some possible directions for investigation in this field and generation of such new models are suggested.