The Schrödinger-Virasoro Lie group and algebra: from geometry to representation theory

Abstract

This article is concerned with an extensive study of an infinite-dimensional Lie algebra $\mathfrak{sv}$, introduced 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 $\mathfrak{sv}$ the Schrödinger-Virasoro algebra. We choose to present $\mathfrak{sv}$ 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 cohomological study, providing in particular a classification of deformations and central extensions; there appears a non-local cocycle.

References in corpus (3)

• Galilean Isometries
• Local scale-invariance and ageing in noisy systems
• On the dynamical symmetric algebra of ageing: Lie structure, representations and Appell systems

Cited by in corpus (6)

• Ternary Virasoro - Witt Algebra
• Representations of the Schrödinger-Virasoro algebras
• The derivation algebra and automorphism group of the twisted Schrödinger-Virasoro algebra
• Classification of Irreducible Weight Modules with a Finite-dimensional Weight Space over the Twisted Schrödinger-Virasoro Lie algebra
• 2-Cocycles of Deformative Schrödinger-Virasoro Algebras
• Leibniz Central Extension on the Twisted Schrödinger-Virasoro Algebra