Foundations of Quantum Groups and Hopf Algebras

Quantum groups represent one of the most fascinating developments in modern mathematical physics, yet they often seem locked behind impenetrable jargon. This course breaks down these complex algebraic structures into clear, accessible concepts for learners looking to expand their mathematical toolkit. By studying the foundational relationships between Lie algebras, Hopf algebras, and their quantum deformations, you will develop a strong intuitive and algebraic grasp of how quantum groups function. What you will learn: Understand the core transition from classical Lie algebras to Hopf algebras; Explore the mechanics of deformation theory and how quantization alters algebraic structures; Analyze the definition and properties of quantized universal enveloping algebras; Practice working with key examples such as the quantum deformation of sl(2); Discover how quantum groups connect to low-dimensional topology and integrable systems. The journey begins with essential algebraic definitions, ensuring you have a solid grasp of bialgebras and Hopf algebras before moving on to deformation and quantization. This text-only course is designed for mathematical beginners and physicists who want a clear introduction to quantum groups without needing advanced prerequisites. Start reading today and build a solid mathematical foundation in quantum symmetries.