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A Darboux-Getzler theorem for scalar difference Hamiltonian operators

Casati, Matteo, Wang, Jing Ping (2019) A Darboux-Getzler theorem for scalar difference Hamiltonian operators. Communications in Mathematical Physics, . ISSN 0010-3616. E-ISSN 1432-0916. (doi:10.1007/s00220-019-03497-2)

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Abstract

In this paper we extend the notion of Poisson-Lichnerowicz cohomology, an object encapsulating the building blocks for the theory of deformations of Hamiltonian operators, to the difference case. A local scalar difference Hamiltonian operator is a polynomial in the shift operator and its inverse, with coefficients in the algebra of difference functions, endowing the space of local functionals with the structure of a Lie algebra. Its Poisson-Lichnerowicz cohomology carries information about the center, the symmetries and the admissible deformations of such an algebra. The analogue notion for the differential case has been widely investigated: the first and most important result is the triviality of all but the lowest cohomology for first order Hamiltonian differential operators, due to Getzler. We study the Poisson-Lichnerowicz cohomology for the operator K0 = S−S^{−1} , which is the normal form for (−1, 1) order scalar difference Hamiltonian operators; we obtain the same result as Getzler did, namely all the cohomology groups except from the 0th e 1st ones vanish. We then apply our main result to the classification of lower order scalar Hamiltonian operators recently obtained by De Sole, Kac, Valeri and Wakimoto.

Item Type: Article
DOI/Identification number: 10.1007/s00220-019-03497-2
Subjects: Q Science > QA Mathematics (inc Computing science)
Q Science > QC Physics > QC20 Mathematical Physics
Divisions: Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User: Matteo Casati
Date Deposited: 27 Jun 2019 15:11 UTC
Last Modified: 02 Aug 2019 14:26 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/73755 (The current URI for this page, for reference purposes)
Casati, Matteo: https://orcid.org/0000-0002-2207-4807
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