Lie algebra is a vector space equipped with a binary operation called the Lie bracket, which measures the failure of the operation to be commutative. Specifically, the Lie bracket of two elements in the vector space is another element in the vector space, and it satisfies certain axioms related to bilinearity, skew-symmetry, and the Jacobi identity.
Lie algebras are named after the Norwegian mathematician Sophus Lie, who first introduced them in the late 19th century as a way of studying the symmetry properties of differential equations. They have since found wide-ranging applications in various areas of mathematics and physics, including group theory, representation theory, geometry, topology, and quantum mechanics.
One of the key features of Lie algebras is that they provide a way of studying Lie groups, which are groups that are also smooth manifolds. The Lie algebra of a Lie group is a vector space that is closely related to the tangent space of the group at the identity element. This connection allows Lie algebras to be used to study the structure and representation theory of Lie groups.
Lie algebras also have important applications in physics, where they provide a framework for studying symmetries and conservation laws in physical systems. For example, the Lie algebra of the Poincaré group, which is the symmetry group of special relativity, gives rise to the concept of angular momentum in quantum mechanics.
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