Deforming the Lie algebra of vector fields on $S^1$ inside the Poisson algebra on $\dot T^*S^1$

Details Deforming the Lie algebra of vector fields on $S^1$ inside the Poisson algebra on $\dot T^*S^1$ Deforming the Lie algebra of vector fields on $S^1$ inside the Poisson algebra on $\dot T^*S^1$

1997-07-07

arxiv.org/abs/q-alg/9707007v1

Mathematics

Quantum Algebra

19 pages, LaTex

Scientific paper

10.1007/s002200050473

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 $\dot T^*S^1$.

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