A Lie group is a mathematical structure that combines algebraic and geometric concepts. It consists of a set of continuous transformations that can be smoothly combined, allowing for operations like multiplication and inversion. Lie groups are essential in various fields, including physics and engineering, as they describe symmetries and conservation laws. The study of Lie groups is closely related to Lie algebras, which are algebraic structures that capture the infinitesimal behavior of the groups. Together, they provide powerful tools for understanding complex systems, such as those found in quantum mechanics and differential equations.