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 −1 2 · . . . · x −1 n · ∂ −1 1 . . . · ∂ −1 n of D ∈ ΨDifn(S 1) (in any coordinate system). It is easy to verify that Tr[D1,D2] = 0 for any D1,D2 ∈ ΨDifn(S 1). Furthermore, lnxi (i = 1, . . . , n) are (exterior) derivations of ΨDifn(S 1) with respect to the adjoint action; the symmetry between the operators x and d dx allows us to define exterior derivations ln ∂i (i = 1, . . . , n). (Actually, one can define lnD (D ∈ ΨDifn(S 1)) in much greater generality — see Sect. 2.4 and 2.8). We prove in §1 that the noncommutative residue Tr on the associative algebra ΨDifn(S 1) and the set of 2n derivations {ln xi, ln ∂i; i = 1, . . . , n} satisfy conditions (i)–(iii) above, and this is our main (and in some sense the unique) example. In the case of the one derivation D such a construction appeared in [KK], where two 2-cocycles on the Lie algebra ΨDif1(S 1) = ΨDif(S1) were constructed: Ψ(A1, A2) = Tr([lnx,A1] ·A2) Ψ(A1, A2) = Tr([ln ∂,A1] ·A2). Both these cocycles are cohomologous to zero when restricted to the Lie algebra Dif1 of the (polynomial) differential operators on C1; on the other hand, our aim is to construct cocycles on this Lie algebra. We are able to accomplish this by the simultaneous application of both lnxi and ln ∂i. Now the problem is solved only for n = 1 (§1) and n = 2 (§3). But the Second Version of the Main Conjecture (§4) gives us an explicit formula for arbitrary n (when conditions (i)–(iii) above hold). We obtain this formulas using a some simple trick, and all that