SO(4)

SO(4)
Name	SO(4)
Type	Lie group
Notation	SO(4)
Connected	yes (component of identity)

SO(4) SO(4) is the group of orientation-preserving isometries of Euclidean 4-space that fix the origin, a compact connected real Lie group of dimension 6 closely related to rotations studied in Isaac Newton, Carl Friedrich Gauss, William Rowan Hamilton, Bernhard Riemann and Sophus Lie. It appears in the work of Élie Cartan, Hermann Weyl, Évariste Galois, Felix Klein and in later formalism by Harish-Chandra and Claude Chevalley. SO(4) plays a role in the research programs of Andrew Wiles, Edward Witten, Michael Atiyah, Isadore Singer, and Roger Penrose.

Definition and basic properties

SO(4) is defined as the subgroup of 4×4 real matrices with determinant 1 preserving the standard Euclidean form, a compact subgroup studied by Carl Ludwig Siegel, Niels Henrik Abel, David Hilbert, Émile Picard and Henri Poincaré. It is a closed subgroup of the general linear group considered by Augustin-Louis Cauchy and Arthur Cayley. The group has finite center and maximal tori analyzed by Élie Joseph Cartan and Hermann Weyl. Its structure underpins work by John Milnor, Raoul Bott, Jean-Pierre Serre, George David Birkhoff and Norbert Wiener.

Lie algebra and algebraic structure

The Lie algebra so(4) consists of real 4×4 skew-symmetric matrices and splits into two ideals analogous to self-dual and anti-self-dual parts investigated in papers by Simon Donaldson, Peter Kronheimer, Charles Fefferman, Enrico Bombieri and Jean Bourgain. Its root system and Cartan subalgebra fit into the classification of semisimple Lie algebras developed by Élie Cartan, Weyl, Kostant and Harish-Chandra. Representation-theoretic analysis was extended by George Mackey, Israel Gelfand, Jacques Tits and Roger Howe. Universal enveloping algebra aspects were treated by G. D. Birkhoff, Nathan Jacobson and Joseph Bernstein.

Group actions and representations

SO(4) acts naturally on R^4, on spheres studied by Henri Poincaré and Jakob Steiner, and on manifolds considered by William Thurston, John Nash, Mikhail Gromov, Shing-Tung Yau and Richard Hamilton. Finite and infinite-dimensional representations relate to work by Harish-Chandra, Weyl, Robert Langlands, Hermann Weyl and Edward Witten. Harmonic analysis on SO(4) connects to projects by Atle Selberg, I. M. Gelfand, Dennis Sullivan, André Weil and John Tate. Induced representations and Frobenius reciprocity echo developments by Frobenius and Issai Schur.

Topology and covering groups

Topologically SO(4) is compact, non-simple, and locally isomorphic to a product of 3-sphere groups treated by William Rowan Hamilton and Arthur Eddington. Its universal cover is related to the group of unit quaternions analyzed by William Rowan Hamilton, Arthur Cayley and Élie Cartan, and covering maps played roles in the studies of Henri Poincaré, Kurt Gödel, John Milnor and René Thom. Fundamental group, homology and cohomology calculations appear in work by Élie Cartan, Jean Leray, Raoul Bott, H. Hopf and Samuel Eilenberg.

Decomposition into SU(2)×SU(2)

A standard isomorphism identifies the double cover of SO(4) with SU(2)×SU(2), building on quaternionic methods of William Rowan Hamilton and later formalized by Élie Cartan, Hermann Weyl, Issai Schur and Évariste Galois. This decomposition underlies studies by Michael Atiyah, Isadore Singer, Simon Donaldson and Andrew Casson. The relation informs instanton constructions used by Alexander Belavin, Vladimir Drinfeld, Mikhail S. Marinov and Philip Candelas and appears in classification efforts by John Conway and Simon Norton.

Applications in geometry and physics

SO(4) appears in classical mechanics texts by Leonhard Euler, Joseph-Louis Lagrange, Pierre-Simon Laplace and in rigid body theory developed further by Sofia Kovalevskaya and Augustin-Louis Cauchy. In general relativity and gauge theory it is central in formulations by Albert Einstein, Hermann Weyl, Abdus Salam, Sheldon Glashow, Steven Weinberg and Gerard 't Hooft. Instantons, monopoles and self-dual solutions exploit SO(4) symmetry in papers by Atiyah, Drinfeld, Hitchin, Manin and Edward Witten. In condensed matter and crystallography SO(4) symmetries occur in analyses by William Lawrence Bragg, Max von Laue and Linus Pauling, and in modern string theory contexts by Edward Witten, Cumrun Vafa, Brian Greene, Joseph Polchinski and Andrew Strominger.

Category:Lie groups