Elements of Lie Theory for Generalized Translation Operators

Abstract

As in the case of topological groups, there are two approaches to the investigation of generalized translation operators—global and infinitesimal. The first one includes the theory of representations, harmonic analysis, the theory of almost periodic functions, etc. For groups, the second approach reduces to the construction of Lie theory, i.e., to the analysis of, generally speaking, nonlinear law of multiplication in a certain neighborhood of the identity element of the Lie group in terms of a linear object, i.e., in terms of the Lie algebra. In the present chapter, we construct elements of Lie theory for generalized translation operators. It is worth noting that the Lie theory of generalized translation operators is far from being complete.