LieTheorie

LieTheorie, or Lie theory, is a branch of mathematics that studies continuous symmetries through Lie groups and their Lie algebras. Named after Sophus Lie, it links algebra, geometry, and analysis and underpins much of modern mathematics and physics. The central objects are Lie groups, smooth manifolds equipped with a group structure compatible with their differentiable structure, and Lie algebras, the tangent space at the identity endowed with a bilinear Lie bracket that encodes infinitesimal symmetries.

Lie groups describe global symmetries; Lie algebras describe infinitesimal ones. The exponential map links them, sending a Lie algebra element to a one-parameter subgroup of the corresponding Lie group. Lie's theorems establish that many problems reduce to the study of Lie algebras and that finite-dimensional Lie algebras can be integrated to Lie groups (up to covering). For connected groups, the Lie algebra often determines the local structure completely, and, in many cases, guides the global topology and representation theory.

The structure theory classifies many Lie algebras: semisimple Lie algebras decompose into simple pieces and are described by root systems, Cartan subalgebras, and the Cartan–Killing classification. This yields the types A_n, B_n, C_n, D_n and the exceptional types E6, E7, E8, F4, G2. Classical Lie groups include GL(n), SL(n), SO(n), and Sp(2n), with corresponding Lie algebras gl(n), sl(n), so(n), sp(2n).

Representation theory studies linear actions of Lie groups and Lie algebras on vector spaces. For compact groups, results like the Peter-Weyl theorem provide a complete picture of unitary representations; highest-weight theory describes finite-dimensional representations of semisimple algebras. Lie theory has wide applications in geometry, analysis, physics (gauge theories, quantum mechanics, relativity), and number theory through harmonic analysis and automorphic forms.