Characteristic classes of flags of foliations and Lie algebra cohomology

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 flag at the origin. The latter encodes characteristic classes of flags 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.

arXiv commentv2: exposition improved, 31 pages, 2 figures

References in corpus (1)

• Hochschild cohomology of the Weyl algebra and traces in deformation quantization

Cited by in corpus (1)

• Stable homology of Lie algebras of derivations and homotopy invariants of wheeled operads