LieGruppen

LieGruppen (Lie groups) are mathematical objects that combine algebraic and geometric structures. A Lie group is a set G that is both a differentiable manifold and a group, such that the multiplication map m: G × G → G and the inversion map i: G → G are smooth. The tangent space at the identity e, denoted g = T_e G, is endowed with a Lie bracket [·,·] arising from the commutator of left-invariant vector fields, making g a Lie algebra, which encodes the local structure of G.

Examples include the additive group R^n, the circle group S^1, the general linear group GL(n,R), the special linear group SL(n,R), the orthogonal group O(n) and the special orthogonal group SO(n), and the unitary groups U(n) and SU(n).

Over the complex numbers, one speaks of complex Lie groups; many properties mirror the real case. Key tools include the exponential map exp: g → G, which relates the Lie algebra to a neighborhood of the identity, and representations, which study actions of Lie groups on vector spaces.

Lie groups play central roles in geometry and physics, especially in describing continuous symmetries and conservation laws. They connect to differential equations, homogeneous spaces, and representation theory. The structure theory distinguishes compact, connected Lie groups and semisimple Lie groups, with a rich classification via root systems and Dynkin diagrams for the simple cases (A_n, B_n, C_n, D_n, E6, E7, E8, F4, G2).

Named after Sophus Lie, their study began in the 19th century and continues to influence modern mathematics and theoretical physics.