Homotopification and categorification of Leibniz conformal algebras

TL;DR

This paper extends Leibniz conformal algebras by introducing homotopy and categorification concepts, establishing new algebraic structures and their relationships through cohomology and equivalences.

Contribution

It introduces $Leib_ abla$-conformal algebras and Leibniz conformal 2-algebras, connecting homotopy, categorification, and cohomology in the context of Leibniz conformal algebras.

Findings

Defined $Leib_ abla$-conformal algebras and characterized their properties.

Introduced Leibniz conformal 2-algebras as categorifications.

Proved the equivalence between Leibniz conformal 2-algebras and 2-term $Leib_ abla$-conformal algebras.

Abstract

Bakalov, Kac and Voronov introduced Leibniz conformal algebras (and their cohomology) as a non-commutative analogue of Lie conformal algebras. Leibniz conformal algebras are closely related to field algebras which are non-skew-symmetric generalizations of vertex algebras. In this paper, we first introduce $Leib_\infty$-conformal algebras (also called strongly homotopy Leibniz conformal algebras) where the Leibniz conformal identity holds up to homotopy. We give some equivalent descriptions of $Leib_\infty$-conformal algebras and characterize some particular classes of $Leib_\infty$-conformal algebras in terms of the cohomology of Leibniz conformal algebras and crossed modules of Leibniz conformal algebras. On the other hand, we also introduce Leibniz conformal $2$-algebras that can be realized as the categorification of Leibniz conformal algebras. Finally, we observe that the category of Leibniz conformal $2$-algebras is equivalent to the category of $2$-term $Leib_\infty$-conformal algebras.