Liegroep

Liegroep, or Lie group, is a mathematical structure that is simultaneously a group and a finite-dimensional smooth manifold, with the group operations (multiplication and taking inverses) being smooth maps. The concept, named after Sophus Lie, was developed to study continuous symmetries in geometry, differential equations, and physics. As a smooth manifold, a Lie group has a well-defined tangent space at every point, and the group structure interacts smoothly with this geometric structure.

Associated to every Lie group is its Lie algebra, defined as the tangent space at the identity element equipped with a Lie bracket arising from the commutator of left-invariant vector fields. The Lie algebra encodes infinitesimal symmetries of the group and determines many global properties. The exponential map exp: g → G sends elements of the Lie algebra to the group, serving as a local diffeomorphism near the identity; for matrix groups it coincides with the matrix exponential.

Common examples include general linear groups GL(n, R), special linear groups SL(n, R), orthogonal groups O(n) and special orthogonal groups SO(n), unitary groups U(n), the circle group S^1, and the additive group R^n. These groups arise naturally in geometry, analysis, and physics, where they describe rotations, translations, and other continuous symmetries.

Lie groups are central to representation theory, differential geometry, and gauge theories in physics. They come with rich structure: connected components, compact or noncompact forms, and corresponding Lie algebras that classify local behavior. The interplay between a Lie group and its Lie algebra, via the exponential map and representations, provides powerful tools for understanding symmetry in mathematics and science.