A systematic way of construction of (2+1)-dimensional dispersionless integrable Hamiltonian systems is presented. The method is based on the so-called central extension procedure and classical R-matrix applied to the Poisson algebras of formal Laurent series. Results are illustrated with the known and new (2+1)-dimensional dispersionless systems.

Abstract. A class of multi-component integrable systems associated to Novikov algebras, which interpolate between KdV and Camassa-Holm type equations, is obtained. The construction is based on the classification of low-dimensional Novikov algebras by Bai and Meng. These multi-component bi-Hamiltonian systems obtained by this construction may be interpreted as Euler equations on the centrally extended Lie algebras associated to the Novikov algebras. The related bilinear forms generating cocycles of first, second and third order are classified. Several examples, including known integrable equations, are presented.

The R-matrix formalism for the construction of integrable systems with infinitely many degrees of freedom is reviewed. Its application to Poisson, noncommutative and loop algebras as well as central extension procedure are presented. The theory is developed for (1 + 1)-dimensional case where the space variable belongs either to R or to various discrete sets. Then, the extension onto (2 + 1)-dimensional case is made, when the second space variable belongs to R. The formalism presented contains many proofs and important details to make it self-contained and complete. The general theory is applied to several infinite dimensional Lie algebras in order to construct both dispersionless and dispersive (soliton) integrable field systems. * Let g be a Lie algebra over the field K of complex or real numbers, K = C or R, that is, g is equipped with a bilinear operation [·, ·] : g × g → g, called a Lie bracket, which is skewsymmetric and satisfies the Jacobi identity. The Lie bracket [·, ·] defines the adjoint action of g on g: ad a b ≡ [a, b]. Definition 2.1 A linear map R : g → g such that the operation [a, b] R := [Ra, b] + [a, Rb] a, b ∈ g (2.1)defines another Lie bracket on g is called the classical R-matrix.