Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 094, 18 pages      arXiv:1906.08388      https://doi.org/10.3842/SIGMA.2019.094

Bi-Hamiltonian Systems in (2+1) and Higher Dimensions Defined by Novikov Algebras

Błażej M. Szablikowski
Faculty of Physics, Division of Mathematical Physics, Adam Mickiewicz University, ul. Uniwersytetu Poznańskiego 2, 61-614 Poznań, Poland

Received June 21, 2019, in final form November 21, 2019; Published online November 29, 2019

Abstract
The results from the article [Strachan I.A.B., Szablikowski B.M., Stud. Appl. Math. 133 (2014), 84-117] are extended over consideration of central extensions allowing the introducing of additional independent variables. Algebraic conditions associated to the first-order central extension with respect to additional independent variables are derived. As result $(2+1)$- and, in principle, higher-dimensional multicomponent bi-Hamiltonian systems are constructed. Necessary classification of the central extensions for low-dimensional Novikov algebras is performed and the theory is illustrated by significant $(2+1)$- and $(3+1)$-dimensional examples.

Key words: Novikov algebras; $(2+1)$- and $(3+1)$-dimensional integrable systems; bi-Hamiltonian structures; central extensions.

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