Vershik, A. M.; Graev, M. I. The Poisson model of the Fock space and representations of current groups. (English. Russian original) Zbl 1256.81058 St. Petersbg. Math. J. 23, No. 3, 459-510 (2012); translation from Algebra Anal. 23, No. 3, 63-136 (2011). Summary: The quasi-Poisson measures are considered, i.e., the \( \sigma \)-finite measures given by a density with respect to a Poisson measure. Representations of current groups are constructed in Hilbert spaces of functionals integrable with respect to a quasi-Poisson measure. For the groups \( O(n,1)\) and \( U(n,1)\), these models give new, more convenient, realizations of the representations in Fock spaces constructed in the previous papers by the authors. A crucial role in considerations is played by spaces of configurations and an analogy between quasi-Poisson and stable measures. Cited in 1 ReviewCited in 3 Documents MSC: 81R10 Infinite-dimensional groups and algebras motivated by physics, including Virasoro, Kac-Moody, \(W\)-algebras and other current algebras and their representations 46C05 Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product) 81V70 Many-body theory; quantum Hall effect 22E70 Applications of Lie groups to the sciences; explicit representations 20C35 Applications of group representations to physics and other areas of science Keywords:current group; integral model; Fock representation; canonical representation; special representation; infinite-dimensional Lebesgue measure PDFBibTeX XMLCite \textit{A. M. Vershik} and \textit{M. I. Graev}, St. Petersbg. Math. J. 23, No. 3, 459--510 (2012; Zbl 1256.81058); translation from Algebra Anal. 23, No. 3, 63--136 (2011) Full Text: DOI References: [1] Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. 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