Foundations of Lie Groups and Lie Algebras

Continuous symmetries shape our understanding of the physical universe, from quantum mechanics to general relativity, and are increasingly vital in modern machine learning. Mastering the mathematics of Lie groups and Lie algebras allows you to systematically analyze these symmetries and their underlying structures. This text-based course guides you step-by-step through the core concepts of Lie theory. You will build a solid intuitive and rigorous foundation, moving from basic matrix groups to representation theory and algebraic decompositions, preparing you to apply these powerful mathematical tools to advanced scientific domains. What you'll learn: - Understand the fundamental relationship between Lie groups and Lie algebras using the exponential map. - Explore classical groups, maximal tori, and structural decompositions such as Cartan and Iwasawa. - Learn the core principles of representation theory for compact and finite groups. - Study the representation-theoretic formulation of physical systems, including the hydrogen atom. - Practice computing root systems, weights, and Lie brackets through guided written exercises. - Discover how Lie theory integrates with modern computational methods and geometric deep learning. You will start with essential definitions of manifolds and matrix groups before moving into algebraic classification and representation theory. This course is ideal for advanced undergraduates, beginning graduate students, and self-taught learners in mathematics, physics, or computer science. Begin your journey into the mathematical language of symmetry today.