Characteristic classes of flags of foliations and Lie algebra cohomology

Transformation Groups. 2015. P. 1–40.

Khoroshkin A.

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.
 

Keywords: foliation, слоение, characteristic classes, Lie algebra cohomology, Когомологии алгебр Ли, Характеристический класс

Publication based on the results of: Теория представлений и математическая физика (2015)