The Schrödinger-Virasoro Algebra: Mathematical structure and dynamical Schrödinger symmetries

✍ Scribed by Jérémie Unterberger, Claude Roger (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 2012
• Tongue: English
• Leaves: 342
• Series: Theoretical and Mathematical Physics
• Edition: 1
• Category: Library

✦ Synopsis

This monograph provides the first up-to-date and self-contained presentation of a recently discovered mathematical structure—the Schrödinger-Virasoro algebra. Just as Poincaré invariance or conformal (Virasoro) invariance play a key rôle in understanding, respectively, elementary particles and two-dimensional equilibrium statistical physics, this algebra of non-relativistic conformal symmetries may be expected to apply itself naturally to the study of some models of non-equilibrium statistical physics, or more specifically in the context of recent developments related to the non-relativistic AdS/CFT correspondence.

The study of the structure of this infinite-dimensional Lie algebra touches upon topics as various as statistical physics, vertex algebras, Poisson geometry, integrable systems and supergeometry as well as representation theory, the cohomology of infinite-dimensional Lie algebras, and the spectral theory of Schrödinger operators.

✦ Table of Contents

Front Matter....Pages i-xlii
Geometric Definitions of $$\mathfrak{s}\mathfrak{v}$$ ....Pages 1-15
Basic Algebraic and Geometric Features....Pages 17-29
Coadjoint Representation of the Schrödinger–Virasoro Group....Pages 31-42
Induced Representations and Verma Modules....Pages 43-55
Coinduced Representations....Pages 57-73
Vertex Representations....Pages 75-123
Cohomology, Extensions and Deformations....Pages 125-145
Action of $$\mathfrak{s}\mathfrak{v}$$ on Schrödinger and Dirac Operators....Pages 147-159
Monodromy of Schrödinger Operators....Pages 161-205
Poisson Structures and Schrödinger Operators....Pages 207-230
Supersymmetric Extensions of the Schrödinger–Virasoro Algebra....Pages 231-272
Back Matter....Pages 273-302

✦ Subjects

Mathematical Methods in Physics;Topological Groups, Lie Groups;Mathematical Physics;Category Theory, Homological Algebra;Statistical Physics, Dynamical Systems and Complexity