The realization of non-transitive Novikov algebras

Novikov algebras were introduced in connection with hydrodynamic-type Poiss on brackets and Hamiltonian operators in the formal variational calculus. W e have given a kind of realization of transitive Novikov algebras through t he Novikov algebras given by S Gelfand and their compatible infinitesimal d eformations in Bai and Meng (2001 J. Phys. A: Math. Gen. 34 3363-72). As a further and continuous study, we extend this realization theory to the nont ransitive Novikov algebras in the paper. In two and three dimensions, we fi nd that all non-transitive Novikov algebras also can be realized as the Nov ikov algebras given by S Gelfand and their compatible infinitesimal deforma tions. Moreover, they have simpler formulae.