On the locality of formal distributions over pre-Lie and Novikov algebras

TL;DR

This paper investigates the locality property of formal distributions over various algebraic structures, demonstrating that the Dong Lemma analogue holds for Novikov algebras but not for pre-Lie and pre-associative algebras.

Contribution

It extends the understanding of the Dong Lemma's applicability to non-Lie algebraic structures, specifically Novikov, pre-Lie, and pre-associative algebras.

Findings

Dong Lemma analogue holds for Novikov algebras.

Dong Lemma analogue does not hold for pre-Lie and pre-associative algebras.

Provides insights into the structure of formal distributions over these algebras.

Abstract

The Dong Lemma in the theory of vertex algebras states that the locality property of formal distributions over a Lie algebra is preserved under the action of a vertex operator. A~similar statement is known for associative algebras. We study local formal distributions over pre-Lie (right-symmetric), pre-associative (dendriform), and Novikov algebras to show that the analogue of the Dong Lemma holds for Novikov algebras but does not hold for pre-Lie and pre-associative ones.