1-Differentiable Deformations of the Poisson Bracket Lie Algebra

Abstract

This chapter and the following will deal with deformations of the Lie algebra structure defined by the Poisson brackets on the space of differentiable functions over a symplectic manifold (general phase space) i.e. with Lie algebra structures on the same space, the bracket of which is ‘close’ (in a sense to be made precise later) to the Poisson bracket. The Poisson bracket itself being a bidifferential operator of order 1 (an operator which acts as a differential operator of order 1 on each function) when acting upon a couple of functions, we shall discuss only those deformations which are given by a series of bidifferential operators. In this first chapter we shall restrict ourselves further, and suppose that these bidifferential operators are of order ⩽1, in which case a complete theory can be developed. We shall begin with a short outline of the Gerstenhaber theory of Lie algebra deformations and its connection with the Chevalley-Eilenberg cohomology of Lie algebras.