For completeness, we recall the following definition from [6]:
An n-cochain (n ∈ ℤ+) of a Lie conformal algebra with coefficients in an -module V is a ℂ-linear map
satisfying the following conditions:
(i)
(conformal antilinearity),
(ii) γ is skew-symmetric with respect to simultaneous permutations of ai's and λi's (skew-symmetry).
As usual, let ⊗0 = ℂ, so that a 0-cochain is an element of V. Denote by
the set of all n-cochains. The differential d of an n-cochain γ is defined as follows:
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where γ is linearly extended over the polynomials in λi. In particular, if γ ∈ V is a 0-cochain, then (dγ)λ(a) = aλγ.
It is known from [6] that the operator d preserves the space of cochains and d2 = 0. Thus the cochains of a Lie conformal algebra with coefficients in its module V form a complex, which is denoted by
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and called the basic complex. Moreover, define a (left) ℂ[∂]-module structure on
by
where ∂V denotes the action of ∂ on V. Then d∂ = ∂d and thus
forms a subcomplex. The quotient complex
is called the reduced complex.
The basic cohomology
of a Lie conformal algebra with coefficients in an -module V is the cohomology of the basic complex
and the (reduced) cohomology
is the cohomology of the reduced complex
.
For a q-cochain
we call γ a q-cocycle if d(γ) = 0; a q-coboundary if there exists a (q — 1)-cochain
such that γ = d(ϕ).
Two cochains γ1 and γ2 are called equivalent if γ1 — γ2 is a coboundary. Denote by
and
the spaces of q-cocycles and q-boundaries, respectively. By Definition 5.2,
The main results of this section are the following.
For the Lie conformal algebra , the following statements hold.
(i) For the trivial module ℂ,
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and
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(ii) If a ≠ 0, then dim Hq (, ℂa) = 0, for q ≥ 0.
(iii) If a ≠ 0, then dim Hq (, MΔ,α) = 0, for q ≥ 0.
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