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Mat. Sb., 1992, Volume 183, Number 2, Pages 52–76 (Mi msb1473)  

This article is cited in 12 scientific papers (total in 12 papers)

Categories of bistochastic measures, and representations of some infinite-dimensional groups

Yu. A. Neretin


Abstract: The following groups are considered: the automorphism group of a Lebesgue measure space (with finite or $\sigma$-finite measure), groups of measurable functions with values in a Lie group, and diffeomorphism groups of manifolds. It turns out that the theory of representations of all these groups is closely related to the theory of representations of some category, which will be called the category of $G$-polymorphisms. Objects of this category are measure spaces, and a morphism from $M$ to $N$ is a probability measure on $M\times N\times G$, where $G$ is a fixed Lie group. For some of the above-mentioned infinite-dimensional groups $\mathfrak{G}$ it is shown that any representation of $\mathfrak{G}$ extends canonically to a representation of some category of $G$-polymorphisms. For automorphism groups of measure spaces this makes it possible to obtain a classification of all unitary representations. Also “new” examples of representations of groups of area-preserving diffeomorphisms of two-dimensional manifolds are constructed.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1993, 75:1, 197–219

Bibliographic databases:

MSC: Primary 22A25, 22E67; Secondary 47D03, 81R10
Received: 08.06.1991

Citation: Yu. A. Neretin, “Categories of bistochastic measures, and representations of some infinite-dimensional groups”, Mat. Sb., 183:2 (1992), 52–76; Russian Acad. Sci. Sb. Math., 75:1 (1993), 197–219

Citation in format AMSBIB
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\by Yu.~A.~Neretin
\paper Categories of bistochastic measures, and representations of some infinite-dimensional groups
\jour Mat. Sb.
\yr 1992
\vol 183
\issue 2
\pages 52--76
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\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1993SbMat..75..197N}
\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1993
\vol 75
\issue 1
\pages 197--219
\crossref{https://doi.org/10.1070/SM1993v075n01ABEH003380}
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. I. Shnirelman, “Generalized fluid flows, their approximation and applications”, GAFA Geom funct anal, 4:5 (1994), 586  crossref  mathscinet  zmath  isi
    2. G. I. Olshanskii, “Weil Representation and Norms of Gaussian Operators”, Funct. Anal. Appl., 28:1 (1994), 42–54  mathnet  crossref  mathscinet  zmath  isi
    3. Yu. A. Neretin, “The group of diffeomorphisms of the half-line, and random Cantor sets”, Sb. Math., 187:6 (1996), 857–868  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. Neretin YA., “Notes on Affine Isometric Actions of Discrete Groups”, Analysis on Infinite-Dimensional Lie Groups and Algebras, ed. Heyer H. Marion J., World Scientific Publ Co Pte Ltd, 1998, 274–320  mathscinet  zmath  isi
    5. Brenier Y., Loeper G., “A Geometric Approximation to the Euler Equations: the Vlasov-Monge-Ampere System”, Geom. Funct. Anal., 14:6 (2004), 1182–1218  crossref  mathscinet  isi
    6. Yu. A. Neretin, “Central extensions of groups of symplectomorphisms”, Mosc. Math. J., 6:4 (2006), 703–729  mathnet  mathscinet  zmath
    7. Yu. A. Neretin, “Sphericity and multiplication of double cosets for infinite-dimensional classical groups”, Funct. Anal. Appl., 45:3 (2011), 225–239  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    8. Yu. A. Neretin, “On the boundary of the group of transformations leaving a measure quasi-invariant”, Sb. Math., 204:8 (2013), 1161–1194  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Brenier Ya., “Remarks on the Minimizing Geodesic Problem in Inviscid Incompressible Fluid Mechanics”, Calc. Var. Partial Differ. Equ., 47:1-2 (2013), 55–64  crossref  mathscinet  zmath  isi
    10. Yu. A. Neretin, “Bi-invariant functions on the group of transformations leaving a measure quasi-invariant”, Sb. Math., 205:9 (2014), 1357–1372  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    11. Yu. A. Neretin, “Infinite symmetric groups and combinatorial constructions of topological field theory type”, Russian Math. Surveys, 70:4 (2015), 715–773  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    12. N. I. Nessonov, “An analogue of Schur–Weyl duality for the unitary group of a $\mathrm{II}_1$-factor”, Sb. Math., 210:3 (2019), 447–472  mathnet  crossref  crossref  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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