Abstract
We gather in this chapter useful algebraic and geometric features of the Schrödinger–Virasoro algebra that have not been derived in the previous chapter because they are not directly related to Newton-Cartan structures. The unifying concept here is that of graduations.
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Notes
- 1.
Vertex representations with integer, resp. half-integer valued G-component indices correspond to the so-called Ramond, resp. Neveu-Schwarz sector.
- 2.
There are four series indexed by n: the Lie algebra Vect(n) of all formal vector fields, and the subalgebras of Vect(n) made up of symplectic, unimodular or contact vector fields.
- 3.
One might also consider the subalgebra of \(\mathcal{A} ({S}^{1})\) defined as \(\mathbb{C}[z,{z}^{-1}] \otimes \mathbb{C}[\partial,{\partial }^{-1}]\); it gives the usual description of the Poisson algebra on the torus \({\mathbb{T}}^{2}\), sometimes denoted SU(∞) [67].
- 4.
Simply recall that this theorem states that there does not exist a common non-trivial extension containing both the Poincaré group and the external gauge group.
- 5.
In many cases, by replacing the quantum fields Ψ(x) by a function ψ(x), one obtains simply an irreducible unitary representation of G on a one-particle state.
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© 2012 Springer-Verlag Berlin Heidelberg
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Unterberger, J., Roger, C. (2012). Basic Algebraic and Geometric Features. In: The Schrödinger-Virasoro Algebra. Theoretical and Mathematical Physics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22717-2_2
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DOI: https://doi.org/10.1007/978-3-642-22717-2_2
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Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22716-5
Online ISBN: 978-3-642-22717-2
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