Lifting formulas, Moyal product, and Feigin spectral sequence

Abstract

It is shown, that each Lifting cocycle ${\Psi }_{2n+1},{\Psi }_{2n+3},{\Psi }_{2n+5},...$ ([Sh1], [Sh2]) on the Lie algebra $\Dif_n$ of polynomial differential operators on an $n$-dimensional complex vector space is the sum of two cocycles, its even and odd part. We study in more details the first case $n=1$. It is shown, that any nontrivial linear combination of two 3-cocycles on the Lie algebra $\Dif_1$, arising from the 3-cocycle~${\Psi }_3$, is not cohomologous to zero, in a contradiction with the Feigin conjecture~[F]. The new conjecture on the cohomology $H^\ndot_\Lie(\Dif_1;\C)$ is made.

The paper has been withdrawn due to a crucial mistake in the arguments

References in corpus (1)

• Extensions and contractions of the Lie algebra of q-pseudodifferential symbols