A lie algebra is a mathematical structure that studies the properties of certain algebraic objects called Lie groups. It consists of a vector space equipped with a binary operation known as the Lie bracket, which satisfies specific properties such as bilinearity, antisymmetry, and the Jacobi identity. Lie algebras are essential in various areas of mathematics and physics, particularly in understanding symmetries and conservation laws. Lie algebras can be classified into different types, such as semisimple and abelian. They play a crucial role in the study of differential equations, quantum mechanics, and the theory of representations. Notable examples include the algebra associated with the rotation group in three-dimensional space, which is fundamental in both classical and quantum physics.