An Elementary Introduction of Integrability in Statistical Physics, Quantum Groups, and Knot Theory

This article provides an elementary introduction to the concept of integrability as it relates to statistical physics, quantum groups, and knot theory. The exploration begins with a detailed examination of the $6$ vertex model, a lattice system in statistical physics. We elucidate how the concept of integrability arises in this model and its crucial connection to the Yang-Baxter equation. The solutions to the Yang-Baxter equation, known as $R$ matrices, form a Yang-Baxter algebra, serving as a foundation for the study of quantum groups. Quantum groups, a fascinating mathematical structure, are introduced as non-commutative, non-co-commutative deformations of enveloping Lie algebras. Their Hopf algebra structure is explored, along with their self-duality properties. We investigate how quantum groups are significant in various mathematical and physical contexts, providing a deeper understanding of their importance.

Delving into the captivating realm of knot theory, we introduce the Jones polynomial, a fundamental algebraic invariant. This polynomial allows us to distinguish between distinct knots, providing valuable information about their underlying structures and properties.

We explore the connections between $R$ matrices and Artin's braid group, revealing how these matrices offer essential representations for the braid group and its interplay with knot theory.

The notion of tangles is introduced, highlighting their role in defining quantum invariants of knots. The concept of a Ribbon category is discussed, showcasing the intricate relationships between tangles, quantum groups, and knot theory.