Introduction to Representations of Lie Groups

Lie groups and their representations are central to modern mathematics and theoretical physics, yet accessing this deep subject can often feel overwhelming. This course provides a clear, structured pathway into the representation theory of both compact and non-compact Lie groups, using clear written explanations and detailed step-by-step mathematical proofs. You will transition from understanding basic topological groups to analyzing complex geometric and algebraic classification theorems. By reading through our structured lessons, you will build a rigorous framework for working with continuous, smooth, and unitary representations. You will learn to bridge the gap between Lie groups and Lie algebras, working confidently with modern tools like D-modules and localization theory. What you'll learn: - Understand foundational definitions of Lie groups, Lie algebras, and continuous representation concepts - Analyze smooth, analytic, and K-finite vectors to bridge infinite-dimensional representations with algebraic structures - Construct and study Harish-Chandra modules and the fundamentals of infinitesimal equivalence - Explore the representation theory of SL(2,R) as a concrete, classic non-compact example - Study advanced algebraic tools including the Harish-Chandra isomorphism and category O - Examine modern geometric methods such as the Borel-Weil theorem and Beilinson-Bernstein localization This text-based course begins with essential algebraic and topological prerequisites, establishing key terminology before moving systematically into advanced classification theory and modern geometric representation techniques. Each section is designed to develop your mathematical intuition through clear, precise proofs and illustrative examples. This course is designed for advanced undergraduate or beginning graduate students in mathematics or theoretical physics who want a self-paced, written introduction to Lie group representations. No advanced prior knowledge of Lie algebras is required, as foundational concepts are introduced at the start. Begin your journey into the elegant structures of representation theory today.