Contraction of a Lie algebra

An operation inverse to deformation of a Lie algebra. Let be a finite-dimensional real Lie algebra, let be its set of structure constants with respect to a fixed basis and let , , be a curve in the group of non-singular linear transformations of such that . Let and let be the structure constants of with respect to the basis . If tends to some limit as , then the algebra defined by these constants relative to the original basis is called a contraction of the initial algebra . The contraction is also a Lie algebra, moreover can be obtained by means of a deformation of . If is the Lie algebra of a Lie group , then the Lie group corresponding to is called a contraction of the group .

Although , in general these algebras are not isomorphic. For example, if , then , so for this contraction the limit algebra is always commutative. The natural generalization of this example is the following: Let be a subalgebra in , let be a subspace complementary to , let, moreover, and for each , for . Then in the limit becomes a commutative ideal of , while at the same time multiplication in and the operation of on remain the same.

In particular, let be the Lorentz group, its Lie algebra and the subalgebra corresponding to the subgroup of rotations of -dimensional space. Then the described contraction of gives the Lie algebra of the Galilean group (see Galilean transformation; Lorentz transformation). Hence the Lorentz algebra is a deformation of the Galilean algebra, and it can be shown that the complexification of the Galilean algebra has no other deformations; in the real case the Galilean algebra can also be a contraction of the orthogonal Lie algebra . An equivalent method of obtaining the Galilean algebra from the Lorentz algebra is to define the Lorentz algebra as the algebra preserving the Minkowski form , and then letting the velocity of light tend to . As long as , the algebra arising is isomorphic to . Analogously, deforming the Poincaré algebra (the inhomogeneous Lorentz algebra), it is possible to obtain the de Sitter algebras and of motions of a space of constant curvature. Correspondingly, setting the curvature to 0, one obtains the Poincaré group as a contraction of the de Sitter group.

The connection between these algebras can be extended to representations. If, as in the described examples, there is a matrix , then each representation of generates a representation of the contraction algebra by the formula

for any . The inverse operation (deformation of a representation) is not possible, in general.

References

[1]	A.O. Barut, R. Raçzka, "Theory of group representations and applications" , 1–2 , PWN (1977)
[2]	E. Inönü, E.P. Wigner, "On the contraction of groups and their representations" Proc. Nat. Acad. Sci. USA , 39 (1953) pp. 510–524
[3]	E.J. Saletan, "Contraction of Lie groups" J. Math. Phys. , 2 (1961) pp. 1–22; 742