This article is devoted to an extensive study of an infinite-dimensional Lie algebra sv, introduced in [14] in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free Schrödinger equation and the central charge-free Virasoro algebra Vect(S 1 ). We call sv the Schrödinger-Virasoro Lie algebra. We study its representation theory: realizations as Lie symmetries of field equations, coadjoint representation, coinduced representations in connection with Cartan's prolongation method (yielding analogues of the tensor density modules for Vect(S 1 )). We also present a detailed cohomological study, providing in particular a classification of deformations and central extensions; there appears a non-local cocycle. IntroductionThere is, in the physical literature of the past decades -without mentioning the pioneering works of Wigner for instance -, a deeply rooted belief that physical systems -macroscopic systems for statistical physicists, quantum particles and fields for high energy physicists -could and should be classified according to which group of symmetries acts on them and how this group acts on them.Let us just point at two very well-known examples: elementary particles on the (3+1)-dimensional Minkowski space-time, and two-dimensional conformal field theory.From the point of view of 'covariant quantization', introduced at the time of Wigner, elementary particles of relativistic quantum mechanics (of positive mass, say) may be described as irreducible unitary representations of the Poincaré group

We study deformations of the standard embedding of the Lie algebra Vect(S 1 ) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle T * S 1 (with respect to the Poisson bracket). We consider two analogous but different problems: (a) formal deformations of the standard embedding of Vect(S 1 ) into the Lie algebra of functions onṪ * S 1 := T * S 1 \S 1 which are Laurent polynomials on fibers, and (b) polynomial deformations of the Vect(S 1 ) subalgebra inside the Lie algebra of formal Laurent series onṪ * S 1 .

We consider a Lie algebra generalizing the Virasoro algebra to the case of two space variables. We study its coadjoint representation and calculate the corresponding Euler equations. In particular, we obtain a bi-Hamiltonian system that leads to an integrable non-linear partial differential equation. This equation is an analogue of the Kadomtsev-Petviashvili (of type B) equation.Mathematics Subject Classification (2000) : 17B68, 17B80, 35Q53