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 TMF, 1997, Volume 113, Number 2, Pages 231–260 (Mi tmf1075)

$P_\infty$ algebra of KP, free fermions and 2-cocycle in the Lie algebra of pseudodifferential operators

P. Winternitza, A. Yu. Orlovb

a Université de Montréal
b P. P. Shirshov institute of Oceanology of RAS

Abstract: The symmetry algebra $P_\infty=W_\infty\oplus H\oplus I_\infty$ of integrable systems is defined. As an example, the classical Sophus Lie point symmetries of all higher KP equations are obtained. It is shown that one (“positive”) half of the point symmetries belongs to the $W_\infty$ symmetries, while the other (“negative”) part belongs to the $I_\infty$ ones. The corresponding action on the tau-function is obtained for the positive part of the symmetries. The negative part can not be obtained from the free fermion algebra. A new embedding of the Virasoro algebra into $\operatorname{gl}(\infty)$ describes conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle.

DOI: https://doi.org/10.4213/tmf1075

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English version:
Theoretical and Mathematical Physics, 1997, 113:2, 1393–1417

Bibliographic databases:

Citation: P. Winternitz, A. Yu. Orlov, “$P_\infty$ algebra of KP, free fermions and 2-cocycle in the Lie algebra of pseudodifferential operators”, TMF, 113:2 (1997), 231–260; Theoret. and Math. Phys., 113:2 (1997), 1393–1417

Citation in format AMSBIB
\Bibitem{WinOrl97} \by P.~Winternitz, A.~Yu.~Orlov \paper $P_\infty$ algebra of KP, free fermions and 2-cocycle in the Lie algebra of pseudodifferential operators \jour TMF \yr 1997 \vol 113 \issue 2 \pages 231--260 \mathnet{http://mi.mathnet.ru/tmf1075} \crossref{https://doi.org/10.4213/tmf1075} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1609042} \transl \jour Theoret. and Math. Phys. \yr 1997 \vol 113 \issue 2 \pages 1393--1417 \crossref{https://doi.org/10.1007/BF02634166} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000072788800002} 

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This publication is cited in the following articles:
1. A. Yu. Orlov, D. M. Shcherbin, “Hypergeometric Solutions of Soliton Equations”, Theoret. and Math. Phys., 128:1 (2001), 906–926
2. Orlov, AY, “Multivariate hypergeometric functions as tau-functions of Toda lattice and Kadomtsev-Petviashvili equation”, Physica D-Nonlinear Phenomena, 152 (2001), 51
3. Nissimov, E, “Symmetries of supersymmetric integrable hierarchies of KP type”, Journal of Mathematical Physics, 43:5 (2002), 2547
4. J. Harnad, A. Yu. Orlov, “Scalar Products of Symmetric Functions and Matrix Integrals”, Theoret. and Math. Phys., 137:3 (2003), 1676–1690
5. Bernal, J, “Soliton-like structures and the connection between the Bq and KP equations”, Chaos Solitons & Fractals, 17:5 (2003), 951
6. Orlov, AY, “Soliton theory, symmetric functions and matrix integrals”, Acta Applicandae Mathematicae, 86:1–2 (2005), 131
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