The Schrödinger-Virasoro Lie group and algebra: from geometry to representation theory
Résumé
This article is concerned with an extensive study of a infinite-dimensional Lie algebra $\goth sv$, 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 $\goth sv$ the Schrödinger-Virasoro Lie algebra. We choose to present $\goth sv$ from a Newtonian geometry point of view first, and then in connection with conformal and Poisson geometry. We turn afterwards to 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 $(S^1)$), and finally Verma modules with a Kac determinant formula. We also present a detailed cohomogical study, providing in particular a classification of deformations and central extensions; there appears a non-local cocycle.