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| connected Lie group | |
|---|---|
| Name | Connected Lie group |
| Type | Mathematical object |
| Field | Élie Cartan, Sophus Lie |
| Examples | Special unitary group, Special orthogonal group, General linear group |
| Related | Lie algebra, Covering space, Representation theory |
connected Lie group
A connected Lie group is a smooth manifold endowed with a compatible group structure that is pathwise connected, arising in the work of Sophus Lie and further developed by Élie Cartan and Hermann Weyl. It combines the differential geometry of manifolds studied in Atlas (mathematics) with the algebraic structure of groups like the Special linear group, Orthogonal group (mathematics), and Unitary group. Connected Lie groups play central roles in the theories of Cartan classification, Representation theory, and the study of continuous symmetries used by Albert Einstein, Isaac Newton, and Niels Bohr in physical models.
A connected Lie group is defined as a group object in the category of smooth manifolds that is non-disconnected, so its underlying manifold is connected and the multiplication and inversion maps are smooth; foundational expositions appear in the work of Élie Cartan, Hermann Weyl, Claude Chevalley, and Armand Borel. Basic properties include that every connected Lie group has an associated Lie algebra (constructed by left-invariant vector fields), a unique maximal connected solvable normal subgroup (the Solvable radical), and a Levi decomposition established by Élie Cartan and formalized by George Mostow. Structural results relate to the existence of maximal compact subgroups proved using techniques from Hodge theory and contributions from Harish-Chandra and Atiyah–Bott.
Classical examples are the connected components of General linear group such as GL(n,R), the Special linear group SL(n,R), the Special orthogonal group SO(n), and the Special unitary group SU(n), as well as solvable groups like the Heisenberg group and nilpotent groups appearing in Morse theory and Weil conjectures. Semisimple connected Lie groups are classified by root systems and Dynkin diagrams developed by Élie Cartan and Wilhelm Killing; famous types include A_n, B_n, C_n, D_n, E_6, E_7, E_8, F_4, G_2, studied by Claude Chevalley, Robert Steinberg, and Armand Borel. Compact connected Lie groups are classified up to finite covers by products of tori and simply connected simple compact groups such as SU(n), Spin(n), and exceptional groups tied to E8 lattice and research by John Conway.
Topological invariants include the fundamental group (studied by Henri Poincaré), higher homotopy groups investigated in work of Hassler Whitney and Jean-Pierre Serre, and cohomology rings computed using methods from Élie Cartan and Alexander Grothendieck tools. Algebraic invariants include the center (examined in classifications by Élie Cartan), maximal tori related to Peter–Weyl theorem contexts, and Weyl groups linked to the studies of Hermann Weyl and Niels Henrik Abel. Characteristic classes for principal bundles over manifolds connect to research by Michael Atiyah and Isadore Singer in index theory and tie into invariants used in the Atiyah–Bott fixed-point theorem.
Every connected Lie group G has an associated finite-dimensional Lie algebra g, a tangent-space algebra at the identity developed in Sophus Lie’s program and formalized by Élie Cartan; the exponential map exp: g → G relates one-parameter subgroups to elements of G and features in proofs by Lazard and Baker–Campbell–Hausdorff formula work with contributions from John Baker and Henri Poincaré. For matrix groups like GL(n,C), SL(n,C), and SO(n), the exponential map is the matrix exponential; for noncompact semisimple groups the behavior of exp and its image involves analysis by Harish-Chandra and structural results from George Mostow.
Connected Lie groups admit simply connected covering groups unique up to isomorphism, constructed via universal covers studied by Henri Poincaré and algebraic topologists like P. A. Smith and Jean Leray. Quotients of simply connected groups by discrete central subgroups yield classical examples such as passing from Spin(n) to SO(n), a transition central to work by Élie Cartan and applications by Paul Dirac. The interplay between the fundamental group and representations is crucial in lifting projective representations, a theme in the research of Eugene Wigner and Bargmann.
Representation theory of connected Lie groups, developed by Hermann Weyl, Harish-Chandra, George Mackey, and I. M. Gelfand, studies continuous homomorphisms into linear groups and unitary representations on Hilbert spaces as in the Peter–Weyl theorem and the Plancherel formula for reductive groups. The unitary dual and tempered representations play central roles in the Langlands program initiated by Robert Langlands and furthered by James Arthur and Harish-Chandra. Induced representations, highest-weight theory, and geometric quantization connect to works by Kirillov, Bertram Kostant, and Bertram Kostant and Alan Weinstein.
Connected Lie groups appear as isometry groups of Riemannian manifolds studied by Bernhard Riemann, symmetry groups in classical mechanics and field theories used by Isaac Newton, James Clerk Maxwell, and Albert Einstein, and gauge groups in the Standard Model refined by Murray Gell-Mann and Sheldon Glashow. They structure holonomy groups in connections examined by Élie Cartan and Berger, moduli problems in algebraic geometry pursued by Alexander Grothendieck, and dualities in string theory developed by Edward Witten and Michael Green. Practical appearances include crystallographic symmetry classifications by Bravais lattice theory and control theory applications influenced by Norbert Wiener.
Category:Lie groups