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, meaning you can move from one transformation to another without any abrupt changes. Lie groups are essential in various fields, including physics and engineering, as they help 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.