Deforming the Lie algebra of vector fields on $S^1$ inside the Lie algebra of pseudodifferential symbols on $S^1$

Abstract

We classify nontrivial deformations of the standard embedding of the Lie algebra $\Vect(S^1)$ of smooth vector fields on the circle, into the Lie algebra~$\PD(S^1)$ of pseudodifferential symbols on $S^1$. This approach leads to deformations of the central charge induced on $\Vect(S^1)$ by the canonical central extension of $\PD(S^1)$. As a result we obtain a quantized version of the second Bernoulli polynomial.