Extensions of Virasoro group and Virasoro algebra by modules of tensor-densities on $S^1$

TL;DR

This paper classifies non-trivial extensions of the diffeomorphism group and vector fields on the circle by tensor-density modules, including analogous results for the Virasoro group and algebra, and also classifies their central extensions.

Contribution

It provides a complete classification of non-trivial and central extensions of the Virasoro group and algebra by tensor-densities, expanding understanding of their structure.

Findings

4 non-trivial extensions of $Diff^+(S^1)$

7 non-trivial extensions of $Vect(S^1)$

Classification of central extensions of these Lie algebras

Abstract

We classify non-trivial (non-central) extensions of the group $Diff^+(S^1)$ of all diffeomorphisms of the circle preserving its orientation and of the Lie algebra $Vect (S^1)$ of vector fields on $S^1$, by the modules of tensor-densities on $S^1$. The result is: 4 non-trivial extensions of $Diff^+(S^1)$ and 7 non-trivial extensions of $Vect (S^1)$. Analogous results hold for the Virasoro group and the Virasoro algebra. We also classify central extensions of constructed Lie algebras. CPT-94/P.3024