Deformations of Vector Fields and Hamiltonian Vector Fields on the Plane

For the Lie algebras L\(H(2)) and L\(W(2)), we study their infinitesimal deformations and the corresponding global ones. We show that, as in the case of L\{W(\)), each integrable infinitesimal deformation of L\(H(2)) and L1(W/(2)) can be represented by a 2-cocycle that defines a global deformation by means of a trivial extension. We also illustrate that all deformations of L\{H{2)) arise as restrictions of deformations of L1(H/(2)). 0. Introduction The investigation of deformations of nilpotent subalgebras L\ (A) of graded infinite-dimensional Lie algebras A is a new area of research. In the well-known work [1], A. Fialowski presents a complete description of the deformations of L\(W(\)), the nilpotent part of the Witt algebra. This and similar works like [2] and [3] show that the properties of deformations of nilpotent parts are completely different from those of the corresponding graded Lie algebra. Usually A is rigid while L\(A) has several nontrivial deformations which have a fairly simple algebraic description. Often the integrable infinitesimal deformations, these are the ones that can be extended to a global deformation, can be represented by a cocycle for which the Lie square equals zero. So, this cocycle defines a global deformation by simply putting the coefficient of tp in the deformed commutator equal to zero for all p > 2. We call this a trivial extension, as it means that the infinitesimal deformation is already a global deformation. Our object is to investigate the deformations of Ll(H(2n)), the nilpotent subalgebra of the Lie algebra of polynomial Hamiltonian vector fields in (2«)dimensional space, and L\(W(m)), the nilpotent subalgebra of the Lie algebra of polynomial vector fields in «i-dimensional space. In this paper, the cases « = 1 and m = 2 are studied. With the help of the Feigin-Fuks spectral sequence we compute infinitesimal deformations of these two Lie algebras. Necessary conditions for their integrability will be derived, and we shall prove that these conditions are also sufficient by constructing global deformations with the prescribed infinitesimal parts. We show that in all cases this can be done by means of a trivial extension, and that these global deformations are unique up Received by the editor March 19, 1993 and, in revised form, December 7, 1993. 1991 Mathematics Subject Classification. Primary 17B56, 17B65, 17B66, 17B68, 17B70.