The Schrödinger-Virasoro Lie Group and Algebra: Representation Theory and Cohomological Study

Abstract

This article is devoted to an extensive study of an infinite-dimensional Lie algebra \(\mathfrak{s}\mathfrak{v}\), introduced in [14] in the context of non-equilibrium statistical physics, containing as subalgebras both the Lie algebra of invariance of the free Schrodinger equation and the central charge-free Virasoro algebra Vect(S1). We call \(\mathfrak{s}\mathfrak{v}\) the Schrodinger-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(S1)). We also present a detailed cohomological study, providing in particular a classification of deformations and central extensions; there appears a non-local cocycle.