Factorizable Representation of Current Algebra —Non commutative extension of the Lévy-Kinchin formula and cohomology of a solvable group with values in a Hilbert Space—

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  • Factorizable Representation of Current Algebra

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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). <BR>The formula for a state functional of a factorizable representation is a non-commutative extension of the Levy-Kinchin formula in probability theory. <BR>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. <BR>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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