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The Poisson model of the Fock space and representations of current groups


Authors: A. M. Vershik and M. I. Graev
Translated by: N. V. Tsilevich
Original publication: Algebra i Analiz, tom 23 (2011), nomer 3.
Journal: St. Petersburg Math. J. 23 (2012), 459-510
MSC (2010): Primary 81R10
Published electronically: March 2, 2012
MathSciNet review: 2896165
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Abstract: 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.


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Additional Information

A. M. Vershik
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: vershik@pdmi.ras.ru

M. I. Graev
Affiliation: Scientific-Research Institute for System Studies, Russian Academy of Sciences, Nakhimovskii Prospekt 36–1, Moscow 117218, Russia
Email: graev_36@mtu-net.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-2012-01204-5
Keywords: Current group, integral model, Fock representation, canonical representation, special representation, infinite-dimensional Lebesgue measure
Received by editor(s): June 29, 2010
Published electronically: March 2, 2012
Additional Notes: Supported by the RFBR grants 08-01-00379a, 09-01-12175-ofi-m (the first author) and 10-01-00041a (the second author).
Article copyright: © Copyright 2012 American Mathematical Society