Segal's contractions, AdS and conformal groups

Abstract

Symmetries and their applications always played an important role in I.E. Segal's work. I shall exemplify this, starting with his correct proof (at the Lie group level) of what physicists call the ``O'Raifeartaigh theorem", continuing with his incidental introduction in 1951 of the (1953) In\"on\"u--Wigner contractions, of which the passage from AdS (SO(2,3)) to Poincar\'e is an important example, and with his interest in conformal groups in the latter part of last century. Since the 60s Flato and I had many fruitful interactions with him around these topics. In a last section I succinctly relate these interests in symmetries with several of ours, especially elementary particles symmetries and deformation quantization, and with an ongoing program combining both.