Lifting formulas II

PDMI PREPRINT — 20/1998 Lifting formulas II

Integrability Formulas. Part II

The terminology and notation used here have been introduced in the following articles: [12], [13], [2], [3], [9], [1], [6], [11], [14], [4], [18], [7], [8], [5], [19], [10], [16], [17], and [15]. For simplicity, we use the following convention: a, x are real numbers, n is an element of N, A is a closed-interval subset of R, f , h, f1, f2 are partial functions from R to R, and Z is an open subse...

Lifting Formulas, Moyal Product, and Feigin Spectral Sequence

It is shown, that each Lifting cocycle Ψ2n+1,Ψ2n+3,Ψ2n+5, . . . ([Sh1], [Sh2]) on the Lie algebra Difn of polynomial differential operators on an n-dimensional complex vector space is the sum of two cocycles, its even and odd part. We study in more details the first case n = 1. It is shown, that any nontrivial linear combination of two 3-cocycles on the Lie algebra Dif1, arising from the 3-cocy...

Cohomology of the Lie Algebras of Differential Operators: Lifting Formulas

(i) Tr(DiA) = 0 for any A ∈ A and any Di ∈ D (ii) [Di,Dj ] = ad(Qij) — inner derivation (Qij ∈ A) for any Di,Dj ∈ D (iii) Alt i,j,k Dk(Qij) = 0 for all i, j, k. The main example of such a situation is the Lie algebra ΨDifn(S 1) of the formal pseudodifferential operators on (S1)n (see [A]). The trace Tr in this example is the “noncommutative residue”, Tr(D) = the coefficient of the term x 1 · x ...

Integration of the Lifting Formulas and the Cyclic Homology of the Algebras of Differential Operators

We integrate the Lifting cocycles Ψ2n+1,Ψ2n+3,Ψ2n+5, . . . ([Sh1], [Sh2]) on the Lie algebra Difn 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 λ on an n-dimensional complex manifold M in the sense of Gelfand–Fuks cohomology [GF] (more precisely, we integrate th...