Integration of the Lifting formulas and the cyclic homology of the algebras of differential operators

Abstract

We integrate the Lifting cocycles ${\Psi }_{2n+1},{\Psi }_{2n+3},{\Psi }_{2n+5},...$ ([Sh1], [Sh2]) on the Lie algebra $\Dif_n$ of holomorphic differential operators on an $n$-dimensional complex vector space to the cocycles on the Lie algebra of holomorphic differential operators on a holomorphic line bundle $\lambda$ on an $n$-dimensional complex manifold $M$ in the sense of Gelfand-Fuks cohomology [GF] (more precisely, we integrate the cocycles on the sheaves of the Lie algebras of finite matrices over the corresponding associative algebras). The main result is the following explicit form of the Feigin-Tsygan theorem [FT1]: $H^\bullet_\Lie(\gl^\fin_\infty(\Dif_n);\C) = \wedge^\bullet(Ψ_{2n+1}, Ψ_{2n+3},Ψ_{2n+5}, ...)$.

24 pages, 2 Postscript figures, LaTeX2e