Hochschild cohomologies of the associative conformal algebra ${\mathrm{C}\mathrm{e}\mathrm{n}\mathrm{d}}_{1,x}$

Abstract

It is established in this work that second Hochshild cohomology group of the associative conformal algebra Cend 1,x is zero. As a corollary, this algebra split off in each extension with a nilpotent kernel.

On the Hochschild Cohomologies of Associative Conformal Algebras with a Finite Faithful Representation

“…Let us summarize Theorem 5, Corollary 5, Example 3, Propositions 2, 3, 4, 5 Corollary 4, and the results of [17,28] to state the ultimate description of those semisimple associative conformal algebras with a FFR that split in every extension with a nilpotent kernel.…”

“…Remark 4. For n = 1, the proof stated above does not work since there is no idempotent e 11 ∈ C. However, it was proved in [28] that H 2 (Cend 1,x , M ) = 0 for every conformal bimodule M over Cend 1,x . As a corollary, Theorem 5 holds also for n = 1.…”

“…, 1, x}, always have trivial second cohomology group. For n = 1, it was done in [28]. In this paper, we use a different method of proof that does not work for n = 1.…”

Morse matching method for conformal cohomologies

“…. It was proved in [16] that H 2 (U(2), M) = 0 for every conformal bimodule over the Weyl conformal algebra U(2) in contrast to 1-dimensional H 2 (Vir, k). This is the reason to study H n (U(3), k) which is the aim of Section 3.…”

“…It was shown in [16] that the second Hochschild cohomology groups H 2 (U(2), M) are trivial for every conformal (bi-)module M.…”