Geodesic
Lie algebra of formal vector fields extended by formal $\mathbf g$-valued functions
Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 19, Tome 335 (2006), pp. 205-230 
A. S. Khoroshkin. Lie algebra of formal vector fields extended by formal $\mathbf g$-valued functions. Zapiski Nauchnykh Seminarov POMI, Questions of quantum field theory and statistical physics. Part 19, Tome 335 (2006), pp. 205-230. http://geodesic.mathdoc.fr/item/ZNSL_2006_335_a10/
@article{ZNSL_2006_335_a10,
     author = {A. S. Khoroshkin},
     title = {Lie algebra of formal vector fields extended by formal $\mathbf g$-valued functions},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {205--230},
     year = {2006},
     volume = {335},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2006_335_a10/}
}
TY  - JOUR
AU  - A. S. Khoroshkin
TI  - Lie algebra of formal vector fields extended by formal $\mathbf g$-valued functions
JO  - Zapiski Nauchnykh Seminarov POMI
PY  - 2006
SP  - 205
EP  - 230
VL  - 335
UR  - http://geodesic.mathdoc.fr/item/ZNSL_2006_335_a10/
LA  - ru
ID  - ZNSL_2006_335_a10
ER  - 
%0 Journal Article
%A A. S. Khoroshkin
%T Lie algebra of formal vector fields extended by formal $\mathbf g$-valued functions
%J Zapiski Nauchnykh Seminarov POMI
%D 2006
%P 205-230
%V 335
%U http://geodesic.mathdoc.fr/item/ZNSL_2006_335_a10/
%G ru
%F ZNSL_2006_335_a10

Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In this work we consider infinite-dimensional Lie-algebra $W_n\ltimes\mathbf g\otimes\mathcal O_n$ of formal vector fields on $n$-dimensional plane, extended by formal $\mathbf g$-valued functions of $n$ variables. Here $\mathbf g$ is an arbitrary Lie algebra. We show that the cochain complex of this Lie algebra is quasi-isomorphic to the quotient of Weyl algebra of $(\mathbf{gl}_n\oplus\mathbf g)$ by $(2n+1)$-st term of standard filtration. We consider separately the case of reductive Lie algebra $\mathbf g$. We show how one can use the methods of formal geometry, to construct characteristic classes of bundles. For every $\mathbf G$-bundle on $n$-dimensional complex manifold we construct a natural homomorphism from ring $A$ of relative cohomologies of Lie algebra $W_n\ltimes \mathbf g\otimes\mathcal O_n$ to ring of tohomologies of the manifold. We show that generators of ring $A$ mapped under this homomorphism to characteristic classes of tangent and $\mathbf G$-bundles.

[1] A. Borel, “Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts”, Annals of Mathematics, 57 (1953), 115–207 | DOI | MR | Zbl

[2] R. Bott, L. V. Tu,, Differentsialnye formy v algebraicheskoi topologii, Platon, Volgograd, 1997

[3] German Veil, Klassicheskie gruppy, ikh invarianty i predstavleniya, IL, M., 1947

[4] I. M. Gelfand, D. A. Kazhdan, D. B. Fuks, “Deistviya beskonechnomernykh algebr Li”, Funkts. analiz, 6:1 (1972), 10–15 | MR | Zbl

[5] V. W. Guillemin, Notes on Gelfand–Fucks. Cohomology, MIT, 1973

[6] I. M. Gelfand, D. B. Fuks, “Kogomologii algebry Li formalnykh vektornykh polei”, Izv. Akad. Nauk SSSR, Ser. Mat., 34:2 (1970), 322–337 | MR | Zbl

[7] A. L. Gorodentsev, A. Khoroshkin, A. N. Rudakov, “On syzygies of highest weight orbits” (to appear)

[8] D. B. Fuks, Kogomologii beskonechnomernykh algebr Li, Nauka, M., 1984 | MR | Zbl

[9] B. Feigin, G. Felder, B. Shoikhet, Hochschild cohomology of the Weyl algebra and traces in deformation quantization, arXiv: /math.QA/0311303 | MR

[10] B. L. Feigin, B. L. Tsygan, “Riemann–Roch theorem and Lie algebra cohomology, 1”, Proc. Winter Sch. Geom. Phys. (Srni, Czech. 1988), Rend. Circ. Mat. Palermo, II. Ser., 21, 1989, 15–52, Suppl | MR | Zbl