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

TL;DR

This review systematically unifies quantum integrable models, both ultralocal and nonultralocal, highlighting their algebraic structures and recent developments in their classification and theory.

Contribution

It provides a comprehensive classification of quantum integrable models based on their algebraic structures, especially emphasizing the unification of ultralocal and nonultralocal models.

Findings

Ultralocal models are classified via quantum algebra and Yang-Baxter equation.

Nonultralocal models are systematized through the braided Yang-Baxter equation.

The paper suggests new directions for research and model generation.

Abstract

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.