Noncommutative Lie algebras

Noncommutative Lie algebras (Leibniz algebras) are defined by identity: [[x, y], z] = [x [y, z]] - [y, [x, z]] Lie algebra of divergenceless vector fields S-2 and Lie algebra of hamiltonian vector fields H-n have noncommutative central extensions (exactly one in each cases). For other Cartan Type Lie algebras any central extension is skew-symmetric. Current algebras are in opposite case: each of them has only one Lie central extension (they are well known as a Kac-Moody algebras).