Historical and Future Development of Lie Group

This article presents a concise historical overview of Lie groups, underscoring their pivotal role in mathematics. It traces the development from Augustin-Louis Cauchy to Hermann Weyl, illustrates the modern definition and integrated features of Lie groups, and highlights Lie groups as essential bridges among mathematical disciplines. This article also introduces applications of Lie groups in mathematics. Lie groups have wide applications in geometry, differential equations, representation theory, and beyond, making them a unifying concept. Lie groups are also very diverse in the future developments, and can be applied in geometry, algebra, and analysis in mathematics, as well as in deep learning in computers, and in nuclear and particle physics.