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arXiv · 1303.1889

Characteristic classes of flags of foliations and Lie algebra cohomology

Anton Khoroshkin
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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.

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Anton Khoroshkin. 2014-08-25. Characteristic classes of flags of foliations and Lie algebra cohomology. https://doi.org/10.1007/s00031-015-9354-5

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