TL;DR
This paper provides a comprehensive review of quantum groups, their mathematical foundations, and their applications in modern physics, emphasizing representation theory and solutions to the Yang-Baxter equation.
Contribution
It offers a systematic introduction to quantum groups via quantum doubles and explores their role in quantum symmetries across physics.
Findings
Analysis of highest weight and cyclic representations at roots of unity
Application of quantum groups to solve the Yang-Baxter equation
Insights into quantum symmetries in modern physics
Abstract
This is a self-contained review on the theory of quantum group and its applications to modern physics. A brief introduction is given to the Yang-Baxter equation in integrable quantum field theory and lattice statistical physics. The quantum group is primarily introduced as a systematic method for solving the Yang-Baxter equation. Quantum group theory is presented within the framework of quantum double through quantizing Lie bi-algebra. Both the highest weight and the cyclic representations are investigated for the quantum group and emphasis is laid on the new features of representations for being a root of unity. Quantum symmetries are explored in selected topics of modern physics.
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Taxonomy
TopicsMolecular spectroscopy and chirality · Algebraic structures and combinatorial models · Spectroscopy and Quantum Chemical Studies