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 ˙T ^* S ¹≔S 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.