A notion of factorizable representation is defined and all factorizable representations of a commutative group of functions as well as those of the current commutation relations and canonical commutation relations are explicitly given in the continuous tensor product space (i.e. the Fock space). The formula for a state functional of a factorizable representation is a non-commutative extension of the Levy-Kinchin formula in probability theory. In the course of analysis, the most general form of a first order cocycle of any solvable group with values in a Hilbert space is determined. Non trivial cohomologies appear by two entirely different mechanism, namely a topological one on infinite dimensional space and an algebraic one on a finite dimensional space. The immaginary part of an inner product of such cocycle is a second order cocycle. The condition that it is a coboundary is discussed.
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Huzihiro Araki, Factorizable Representation of Current Algebra. Publ. Res. Inst. Math. Sci. 5 (1969), no. 3, pp. 361–422DOI 10.2977/PRIMS/1195194390