symmetry in nLab

Contents

Contents

Idea

A symmetry is (in most classical situations) invariance under a group action, or infinitesimally, invariance under a Lie algebra action. Indeed, historically the mathematical term “group” is a contraction of group of symmetries (e.g. Klein 1872). More recently,the concept of symmetry has been a popular topic of study in the Physics literature under the name of generalized symmetry.

In physics, see local symmetry, global symmetry, asymptotic symmetry spontaneous symmetry breaking.

Examples

• Euclidean group, space group/crystallographic group

• Lorentz group

• …

• automorphism group

• invariant, invariant theory

• stabilizer subgroup, Erlangen program

• symmetry of a Lagrangian

• quantum symmetry

• supersymmetry

References

Historical articles

• Felix Klein, Vergleichende Betrachtungen über neuere geometrische Forschungen (1872)

  translation by M. W. Haskell, A comparative review of recent researches in geometry , trans. M. W. Haskell, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (retyped pdf, retyped pdf, scan of original)

  (Erlanger program)

• Hermann Weyl, Symmetry, Journal of the Washington Academy of Sciences 28 6 (1938) 253-271 [jstor:24530200]

On transformation groups:

• Willard Miller, Symmetry Groups and Their Applications, Pure and Applied Mathematics 50 (1972) 16-60 [ISBN:9780080873657]

On symmetry and introducing the language of homotopy type theory for univalent foundations of mathematics:

• Marc Bezem, Ulrik Buchholtz, Pierre Cagne, Bjørn Ian Dundas, Daniel R. Grayson: Symmetry (2021) [[pdf]]