Deforming the Lie algebra of vector fields on S1S^1 inside the Poisson algebra on T˙∗S1\dot T^*S^1

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 $\dot T^*S^1$.Comment: 19 pages, LaTe

Conformally invariant differential operators on tensor densities

Let ${\cal F}_\lambda$ be the space of tensor densities on ${\bf R}^n$ of degree $\lambda$ (or, equivalently, of conformal densities of degree $-\lambda{}n$) considered as a module over the Lie algebra $so(p+1,q+1)$. We classify $so(p+1,q+1)$-invariant bilinear differential operators from ${\cal F}_\lambda\otimes{\cal F}_\mu$ to~${\cal F}_\nu$. The classification of linear $so(p+1,q+1)$-invariant differential operators from ${\cal F}_\lambda$ to ${\cal F}_\mu$ already known in the literature is obtained in a different manner.Comment: 11 pages, LaTe