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

Abstract

Abstract. It is proved that the second cohomology group of the conformal algebras Cend n and Cur n with coefficients in any bimodule is trivial. As a result, these algebras are segregated in any 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.…”

“…[17]). One has H 2 (Cend n , M ) = 0 for every conformal bimodule M over the conformal algebra Cend n .…”

$\mathcal{O}$-operators and Nijenhius operators of associative conformal algebras

“…Hence, if we look for a new λ -product • λ which is compatible in the sense of (5.12), then this means that the new λ -product • λ is a 2-cocycle of the original associative conformal algebra. If our algebra is, for instance, the current conformal algebra Cur n or the conformal algebra Cend n , it is proved in [22] that the second cohomology group of Cend n and Cur n with coefficients in any conformal bimodule is trivial, hence our λ -product • λ has to be a coboundary, namely, of the form • N λ for some N. This means that we have not much freedom and, looking for compatible associative λ -products, we must, in principle, work with Nijenhuis operators.…”

Hochschild cohomologies of the associative conformal algebra Cend 1,x

“…The radical splitting problem of this class of algebras was considered in [11, see also 1,2]. There is a definition of Hochschild cohomology for associative conformal algebras in [1,2] and in a context of pseudoalgebras [3] (slightly different from the definition in [4]). Cohomology play a key role in the radical splitting problem.…”