Abstract : We discuss the general problem of symmetry beakdown in the algebraic approach to statistical mechanics. We consider in particular the case of classical quantum lattice systems.]]>

Abstract : We present S. Lie's work on the symmetry of ODE (ordinary differential equations) : action of Diff 2(x,y), the Lie algebra of vector fields of the plane x, y, on the set of ODE ; general form of ODE with a given symmetry algebra ; computation of the symmetry algebra of a given equation. The original part of this conference studies the general linear equation of order n>2 (this has also be done independenty by Mohamed and Leach, ref. [9]). We also present in a more precise form the work of Lie on the finite dimensional subalgebras of Diff2. As an application, we classify the symmetries of second order equations.]]>