Abstract
Suppose, that we have selected a certain non-compact group G as the higher symmetry group of some physical system. Then to regard all consequences and restrictions which follow from the introduced symmetry we should examine the following questions: 1. What is the system of invariant operators which generate the ring of invariant operators? Have we the possibility of a reduction of this system if we have no convenient interpretation for these operators as physical observables? 2. What is the maximal system of commuting operators in the fixed representation space which we shall interpret as physical observables? What is the shape and range of their spectra? 3. What are the properties of the basic functions in the representation space, which we would like to identify with physical states? 4. What are the properties of the direct product of two representations (Clebsch-Gordan series) and two basic vectors (Clebsch-Gordan coefficients)? 5. What are the properties of the decomposition of a considered representation with respect to a maximal compact subgroup (which is often an initial symmetry group)?
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