Deforming the Lie Algebra of Vector Fields on $S^1$ Inside the Poisson Algebra on ˙ T * S 1

Abstract

We study deformations of the standard embedding of the Lie algebra $\Vect(S^1)$ of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle $T^*S^1$ (with respect to the Poisson bracket). We consider two analogous but different problems: (a) formal deformations of the standard embedding of $\Vect(S^1)$ into the Lie algebra of functions on $\dot T^*S^1:=T^*S^1\setminusS^1$ which are Laurent polynomials on fibers, and (b) polynomial deformations of the $\Vect(S^1)$ subalgebra inside the Lie algebra of formal Laurent series on $\dot T^*S^1$.