In theoretical physics, Eugene Wigner and Erdal İnönü have discussed the possibility to obtain from a given

Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...

a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a limiting operation on a parameter of the

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...

, altering the

structure constant In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination. Given the structure constants, the resulting pr ...

s of this Lie algebra in a nontrivial singular manner, under suitable circumstances. For example, the Lie algebra of the

3D rotation group In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space \R^3 under the operation of composition. By definition, a rotation about the origin is a tr ...

, , etc., may be rewritten by a change of variables , , , as : . The contraction limit trivializes the first commutator and thus yields the non-isomorphic algebra of the plane

Euclidean group In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). ...

, . (This is isomorphic to the cylindrical group, describing motions of a point on the surface of a cylinder. It is the little group, or

stabilizer subgroup In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...

, of null four-vectors in Minkowski space.) Specifically, the translation generators , now generate the Abelian

normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G i ...

of (cf. Group extension), the parabolic Lorentz transformations. Similar limits, of considerable application in physics (cf.

Correspondence principle In physics, the correspondence principle states that the behavior of systems described by the theory of quantum mechanics (or by the old quantum theory) reproduces classical physics in the limit of large quantum numbers. In other words, it says t ...

s), contract * the de Sitter group to the Poincaré group , as the de Sitter radius diverges: ; or * the Poincaré group to the

Galilei group In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotatio ...

, as the speed of light diverges: ; or * the

Moyal bracket In physics, the Moyal bracket is the suitably normalized antisymmetrization of the phase-space star product. The Moyal bracket was developed in about 1940 by José Enrique Moyal, but Moyal only succeeded in publishing his work in 1949 after a le ...

Lie algebra (equivalent to quantum commutators) to the

Poisson bracket In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. Th ...

Lie algebra, in the classical limit as the Planck constant vanishes: .

Notes

References

* * * * * {{Cite journal, last1=Segal, first1=I. E., author-link = Irving Segal, doi = 10.1215/S0012-7094-51-01817-0, title=A class of operator algebras which are determined by groups, journal=

Duke Mathematical Journal ''Duke Mathematical Journal'' is a peer-reviewed mathematics journal published by Duke University Press. It was established in 1935. The founding editors-in-chief were David Widder, Arthur Coble, and Joseph Miller Thomas Joseph Miller Thomas (16 ...

, volume=18, pages=221, year=1951 Lie algebras Lie groups Mathematical physics