On the central extensions of Poisson brackets of hydrodynamic type

Abstract

Poisson brackets of hydrodynamic type were introduced to construct a theory of conservation system of hydrodynamic type. They lead to the algebraic and differentially geometrical theory of local translationally invariant Lie algebras of first order, Frobenius-type algebras and Novikov algebras. One of the main ways to construct new examples of these algebras and their corresponding Poisson brackets is through the central extensions. In Balinskii A A and Novikov S P (1985 Sov. Math. Dokl. 32 228–31), there is a general theory of the central extensions. In this paper, we give a further detailed study of two (non-trivial) types of Novikov algebras which directly decide an important kind of central extension of the Poisson brackets. We find that the Novikov algebras in the case τ = 3 are in the case τ = 0 and these two cases coincide in dimension ≤ 3. Moreover, they also coincide for the transitive cases in dimension 4.