On Koszulity for operads of Conformal Field Theory

TL;DR

This paper proves the Koszulity of the Conformal Lie operad using operadic Groebner bases, explores deformations of operads, and introduces Hom-Gelfand-Dorfman algebras, advancing the understanding of algebraic structures in conformal field theory.

Contribution

It establishes the Koszulity of the Conformal Lie operad and introduces Hom-Gelfand-Dorfman algebras, providing new tools for studying operadic deformations in conformal field theory.

Findings

Proved Koszulity of the Conformal Lie operad.

Identified a unique confluent deformation of the Associative operad.

Introduced and studied properties of Hom-Gelfand-Dorfman algebras.

Abstract

We study two closely related operads: the Gelfand-Dorfman operad GD and the Conformal Lie Operad CLie. The latter is the operad governing the Lie conformal algebra structure. We prove Koszulity of the Conformal Lie operad using the Groebner bases theory for operads and an operadic analogue of the Priddy criterion. An example of deformation of an operad coming from the Hom structures is considered. In particular we study possible deformations of the Associative operad from the point of view of the confluence property. Only one deformation, the operad which governs the identity $(\alpha(ab))c=a(\alpha (bc))$ turns out to be confluent. We introduce a new Hom structure, namely Hom--Gelfand-Dorfman algebras and study their basic properties.