## Abstract

We prove the conjecture by Feigin, Fuchs, and Gelfand describing the Lie algebra cohomology of formal vector fields on an n-dimensional space with coefficients in symmetric powers of the coadjoint representation. We also compute the cohomology of the Lie algebra of formal vector fields that preserve a given ag at the origin. The latter encodes characteristic classes of ags of foliations and was used in the formulation of the local Riemann-Roch Theorem by Feigin and Tsygan. Feigin, Fuchs, and Gelfand described the first symmetric power and to do this they had to make use of a fearsomely complicated computation in invariant theory. By the application of degeneration theorems of appropriate Hochschild-Serre spectral sequences, we avoid the need to use the methods of FFG, and moreover, we are able to describe all the symmetric powers at once.

Original language | English |
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Pages (from-to) | 479-518 |

Number of pages | 40 |

Journal | Transformation Groups |

Volume | 21 |

Issue number | 2 |

DOIs | |

State | Published - 1 Jun 2016 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2015, Springer Science+Business Media New York.

## ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology