Cohomology of Conformal Algebras

Hochschild cohomologies for associative conformal algebras

“…In [4], another definition of cohomologies is couched: it uses (· λ ·)-products. That definition, in our opinion, is less appropriate than ours, for it does not give a clear idea as to how cohomologies are related to extensions of a conformal algebra A.…”

Classification of quasifinite modules over Lie algebras of matrix differential operators on the circle

“…Since V is irreducible, we have V = V (α) for some α ∈ C (if we have used the isomorphism σ in (2.8) in the above proof, then V is the module V (α)). Following the proof of Lemma 3.2 (now G m is not necessarily diagonal), we have U = U (1) ⊕U (2) , and both U (1) and U (2) are the natural gl N -modules. Since D(U ) ⊂ U , [D, gl N ] = 0 and V is not irreducible, the subspace U = {u ∈ U | Du ∈ Cu} of eigenvectors of D is a proper (and thus simple) gl N -submodule of U (isomorphic to C N as a gl N -module) and D| U is a scalar map λ for some λ ∈ C. Thus U = U ⊕U , where U is another copy of U such that Du = λu +u for u ∈ U , where u ∈ U is the corresponding copy of u .…”

“…There is a one-to-one correspondence between Lie conformal algebras and maximal formal distribution Lie algebras [1,6,7]. The Lie algebra D N of N × N -matrix differential operators on the circle is a formal distribution Lie algebra associated to the general Lie conformal algebra gc N .…”

“…For the general N , the classification was obtained by Boyallian, Kac, Liberati and Yan [2]. In [13], the author obtained the classification of the irreducible quasifinite modules and indecomposable uniformly bounded modules over D 1 . We would like to take this chance to point out that there is a slight gap in the proof of Proposition 2.2 of [13]; this gap has been filled in this paper (see Remark 3.4).…”

Triviality of the second cohomology group of the conformal algebras $\mathrm {Cend}_n$ and $\mathrm {Cur}_n$

“…The notion of Hochschild cohomology of a conformal algebra was given in [7] by using the so-called λ-product. However, it is not clear how to establish the correspondence between cohomology and associative conformal algebra extensions.…”