The introduction in physics of a finite geometry approximating the ordinary Euclidean one poses the problem of studying the relativity groups over such a geometry. We present a detailed analysis of the structure and irreducible representations of the rotation, Lorentz, and Poincare groups. It is found that, besides the usual quantum numbers, a new two‐valued label is necessary to specify the representations.
Finite Lorentz groups acting on four-dimensional vector spaces coordinatized by finite fields with a prime number of elements are represented as homomorphic images of countable, rational subgroups of the Lorentz group acting on real four-dimensional space-time. Bounded subsets of the real Lorentz group are retractable with arbitrary precision to finite subsets of such rational subgroups. These finite retracts correspond, via local isomorphisms, to well-behaved subsets of Lorentz groups over finite fields. This establishes a relationship of approximation between the real Lorentz group and Lorentz groups over very large finite fields.
This is a brief overview of researches of Dmitry Petrovich Zhelobenko (1934--2006). He is the best known for his book"Compact Lie groups and their representations"and for the classification of all irreducible representations of complex semisimple Lie groups. We tell also on other his works, especially on the spectral analysis of representations.