Contraction of Lie Groups

Abstract

Contraction is defined for a Lie group to coincide on its Lie algebra with a generalization of contraction as first introduced by Inönü and Wigner. This is accomplished with a sequence of nonsingular coordinate transformations on the group (or nonsingular linear coordinate transformations on its Lie algebra), whose limit is a singular one. Essentially all of the calculations are performed in the algebra. It is assumed that in the limit the association in the algebra (the multiplication law in the group) converges, and this gives a necessary and sufficient condition on the algebra. Once it is satisfied, the new (contracted) algebra is uniquely determined in terms of the original one. It is found that the contracted algebra can be further contracted in the same way, and likewise the algebra so obtained. In this way one obtains a terminating sequence of algebras. Inönü-Wigner contraction corresponds to a sequence terminating at the first contraction. Some properties of the original and contracted algebras are studied, and some specific examples are given. Contraction of a Lie algebra induces contraction of any of its representations. This is examined for the case of finite-dimensional representations. Ray representations are discussed in general. It is shown how the trivial exponent of the Lorentz group changes under contraction to the nontrivial one of the Galilei group.