On the Hochschild cohomologies of associative conformal algebras with a finite faithful representation

TL;DR

This paper characterizes semisimple associative conformal algebras with finite faithful representations that have trivial second Hochschild cohomology, providing insights into their structure and radical splitting.

Contribution

It offers a complete classification of semisimple associative conformal algebras with trivial Hochschild cohomology and solves the radical splitting problem for algebras with finite faithful representations.

Findings

Identified all semisimple conformal endomorphism algebras with trivial Hochschild cohomology.

Provided a complete solution to the radical splitting problem in this class.

Enhanced understanding of the structure of associative conformal algebras with finite faithful representations.

Abstract

Associative conformal algebras of conformal endomorphisms are of essential importance for the study of finite representations of conformal Lie algebras (Lie vertex algebras). We describe all semisimple algebras of conformal endomorphisms which have the trivial second Hochschild cohomology group with coefficients in every conformal bimodule. As a consequence, we state a complete solution of the radical splitting problem in the class of associative conformal algebras with a finite faithful representation.