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-cocycle Ψ3, is not cohomologous to zero, in a contradiction with the Feigin conjecture [F]. The new conjecture on the cohomology H• Lie(Dif1;C) is made. Introduction The cocycles Ψ2n+1,Ψ2n+3,Ψ2n+5, . . . on the Lie algebra gl fin ∞ (Difn) of finite matrices over polynomial differential operators on an n-dimensional complex vector space, were constructed in the author’s works [Sh1], [Sh2] (the lower index denotes the degree of the cocycle). These cocycles are called Lifting cocycles. It was proved in [Sh3] that H Lie(gl fin ∞ (Difn);C) = ∧ (Ψ2n+1,Ψ2n+3,Ψ2n+5, . . . ). The situation is much more complicated for the pull-backs of the Lifting cocycles under the map of the Lie algebras Difn →֒ gl fin ∞ (Difn), D 7→ E11 ⊗D. For example, it is known that the first Lifting cocycle Ψ2n+1 on the Lie algebra Difn is not cohomologous to zero, but the same problem for the higher cocycles Ψ2n+2l+1, l > 0 is open, even in the simplest case n = 1. In the present paper we prove that there exists the decomposition of each Lifting cocycle on the Lie algebra Difn in the sum of two cocycles, its even and odd part, Ψ2k+1 = Ψ even 2k+1 + Ψ odd 2k+1. To explain this fact, we work with the Moyal star-product on C[p, q] instead of the usual product in the algebra of differential operators (in the case n = 1). In the terms of the algebra Dif1 it means, that we work with a nonstanard basis in Dif1, which can be obtained from the usual basis {p q} in C[p, q] from the Poincaré–Birkhoff–Witt map. The reason is that in the Moyal product f ∗ g = f · g + ħB1(f, g) + ħ B2(f, g) + . . . (f, g ∈ C[p, q]) the bidifferential operators Bi(f, g) are skew-symmetric on f and g for odd i and are symmetric on f and g for even i. In particular, [f, g] = f ∗ g − g ∗ f = 2(ħB1(f, g) + ħ B3(f, g) + ħ B5(f, g) + . . . ) contains only the bidifferential operators B2i+1, i ≥ 0. This simple trick leads us directly to the decomposition Ψ2k+1 = Ψ even 2k+1 + Ψ odd 2k+1. In the case n = 1 we prove that the