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
T
˙
∗
S
1
.