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| Orthogonal group (mathematics) | |
|---|---|
| Name | Orthogonal group |
| Type | Group, Lie group |
| Field | Mathematics |
| Related | Special orthogonal group, Unitary group, Symplectic group |
Orthogonal group (mathematics) The orthogonal group arises as the group of linear isometries preserving a nondegenerate quadratic form on an n-dimensional real vector space. It is a fundamental example in the study of symmetry appearing across algebra, geometry, and mathematical physics, connecting to classical figures and institutions such as Euclid, Isaac Newton, Carl Friedrich Gauss, Bernhard Riemann, and modern programs at Institute for Advanced Study and École Normale Supérieure.
For a real vector space V of dimension n equipped with a nondegenerate quadratic form Q, the orthogonal group consists of linear automorphisms g with Q(gv)=Q(v) for all v; when Q is the standard form this group is denoted O(n). Foundational results connect O(n) to work of Galois, Évariste Galois, William Rowan Hamilton, and Arthur Cayley through invariant theory and classical algebra. Key algebraic properties include being a closed subgroup of the general linear group GL(n,R) and having two connected components distinguished by determinant ±1, a fact related to studies by Niels Henrik Abel and Élie Cartan. Over other fields such as finite fields studied by Emil Artin and André Weil, orthogonal groups exhibit distinct behavior and feed into the theory of Chevalley groups and Tate conjecture contexts.
With respect to an orthonormal basis for the standard positive-definite form, elements of O(n) are real n×n matrices A satisfying A^T A = I. Classical examples include O(1)≈{±1}, O(2) containing rotations and reflections related to work in Leonhard Euler's studies of rigid motion, and O(3) central to Isaac Newton's mechanics and James Clerk Maxwell's electromagnetism. Noncompact analogues such as O(p,q) for indefinite forms appear in research by Hermann Minkowski, Albert Einstein, and Hermann Weyl in connection with spacetime symmetries and Lorentz transformations. Matrix models tie to computational projects at institutions like Bell Labs and Los Alamos National Laboratory via algorithms for orthogonalization attributed to Carl Friedrich Gauss and John von Neumann.
As a closed subgroup of GL(n,R), O(n) is a compact real manifold with topology and homotopy type investigated by Henri Poincaré and Emmy Noether. Its connected component SO(n) has the same dimension n(n−1)/2 and appears in classical topology studies linked to Poincaré conjecture precursors and work by René Thom and Raoul Bott. Homotopy groups of O(n) and Bott periodicity, established by Raoul Bott and advanced by Michael Atiyah and Isadore Singer, play central roles in K-theory developed at University of Oxford and Massachusetts Institute of Technology.
O(n) is a real Lie group whose Lie algebra consists of skew-symmetric matrices so(n); the exponential map links so(n) to one-parameter subgroups as in classical studies by Sophie Germain and Augustin-Louis Cauchy. The representation theory and structure theory of so(n) align with contributions by Élie Cartan, Hermann Weyl, Harish-Chandra and inform modern classification results by Claude Chevalley and George Lusztig. Connections to root systems and Dynkin diagrams tie O(n) to the Cartan-Killing classification used at places like Princeton University and University of Cambridge.
Important subgroups include the special orthogonal group SO(n), the orthogonal similitude group GO(n), and maximal tori studied in the work of Élie Cartan and Hermann Weyl. Reflection subgroups relate to Coxeter groups investigated by H.S.M. Coxeter and to crystallographic groups of Camille Jordan and A.A. Bravais in solid state contexts. Over finite fields, orthogonal groups are central examples in the classification of finite simple groups involving researchers at University of Chicago and Institute for Advanced Study such as Daniel Gorenstein.
Representation theory of O(n) and SO(n) yields classical invariants like determinants, Pfaffians, and characteristic classes central to studies by David Hilbert, Emmy Noether, Élie Cartan, and Shiing-Shen Chern. Tensor representations, Young tableaux techniques developed by Alfred Young, and branching rules connect to symmetric group work by Frobenius and Issai Schur. Invariant theory for O(n) interacts with algebraic geometry programs at Harvard University and École Polytechnique, influencing modern treatments in the hands of Jean-Pierre Serre and Alexander Grothendieck.
Orthogonal groups appear in classical mechanics and relativity through Isaac Newton's rigid body theory and Albert Einstein's Lorentz group, in quantum mechanics shaped at Cavendish Laboratory and Los Alamos National Laboratory, and in gauge theory and topology studied by Michael Atiyah and Edward Witten. In number theory and arithmetic geometry, orthogonal groups underpin theta correspondence explored by André Weil and automorphic forms research at Institute for Advanced Study and Clay Mathematics Institute. Applications extend to crystallography from A.A. Bravais's lattice work, to robotics and computer vision developed at Stanford University and Carnegie Mellon University, and to coding theory and cryptography influenced by Claude Shannon and Alan Turing.
Category:Lie groups Category:Linear algebra