Deforming the Lie Algebra of Vector Fields on S(1) Inside the Poisson Algebra on ˙T*S1

We study deformations of the standard embedding of the Lie algebra Vect(S¹) of smooth vector fields on the circle, into the Lie algebra of functions on the cotangent bundle T^*S¹ (with respect to the Poisson bracket). We consider two analogous but different problems: (a) formal deformations of the standard embedding of Vect(S¹) into the Lie algebra of functions on ˙T^*S¹≔S¹ which are Laurent polynomials on fibers, and (b) polynomial deformations of the Vect(S¹) subalgebra inside the Lie algebra of formal Laurent series on ˙T^*S¹.