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

## TL;DR

This paper extends the theory of bi-Hamiltonian systems derived from Novikov algebras to higher dimensions by exploring central extensions, providing classifications and concrete examples in (2+1) and (3+1) dimensions.

## Contribution

It introduces algebraic conditions for central extensions in higher-dimensional bi-Hamiltonian systems based on Novikov algebras, expanding previous work to include additional independent variables.

## Key findings

- Constructed higher-dimensional multicomponent bi-Hamiltonian systems.
- Derived algebraic conditions for first-order central extensions.
- Classified low-dimensional Novikov algebras and provided explicit examples.

## 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.