The Galilei group as contraction of the Poincare group

The group of symmetries of special relativity is the Poincare group.

However, before Einstein invented the theory of relativity, physics was believed to follow Newton's laws, and these have a different group of symmetries - the Galilei group, and its infinitesimal symmetries form the Galilei algebra.

Now Newton's physics is just a special case of the theory of relativity in which all motions are very slow compared to the speed of light. Physicists speak of the 'nonrelativisitic limit'. Thus one would expect that the Galilei group is a kind of nonrelativistic limit of the Poincar'e group.

This notion has been made precise by Inonu. He looked at the Poincar'e algebra and 'contracted' it in an ingenious way to the Galilei algebra. The construction could then be lifted to the corresponding groups. Not only that, it turned out to be a general machinery applicable to all Lie algebras and Lie groups, and therefore has found many applications far beyond that for which it was originally developed.