About the Book: This book provides a clear, concise introduction to compact connected Lie groups and their representations, emphasizing both algebraic and geometric viewpoints. It presents the structure and classification theory systematically, using a non-traditional organization that balances rigorous proofs with conceptual understanding. Key topics include the orbital geometry of the adjoint action, which captures the non-commutativity of a Lie group, as well as É. Cartan’s maximal tori theorem and the Weyl group reduction, which clarify the structure and symmetries of these groups. The text is designed to give readers both technical mastery and geometric intuition.