Differential Lie Coalgebras and Lie Conformal Algebras
Carina Boyallian, Jose I. Liberati
TL;DR
This work extends Michaelis-type dualities to the conformal setting by introducing the upper zero functor $(-)^0$ from Lie conformal algebras to differential Lie coalgebras and establishing its adjunction with the conformal dual $(-)^{*c}$. It also defines the Loc functor to extract locally finite subcoalgebras and shows that for free finite-rank Lie conformal algebras, Loc$(L^{0})$ consists of conformal functionals whose kernels contain a cofinite-rank ideal, though $L^{0}$ need not be locally finite. The authors provide explicit examples demonstrating that Loc$(L^{0})$ can be strictly smaller than $L^{0}$, revealing subtleties in local finiteness, and extend the framework to Jordan conformal algebras and differential Jordan coalgebras, preserving the dualities and locality phenomena. Together, these constructions furnish a robust categorical toolkit for dualizing conformal Lie structures and their Jordan analogues, with implications for finite-rank classifications and the interplay between conformal algebras, coalgebras, and their locally finite parts.
Abstract
We define a functor from the category of Lie conformal algebras to the category of differential Lie coalgebras, which associates to any Lie conformal algebra $L$ a differential Lie coalgebra $L^{\,0}$, defined as the maximal good $\mathbb{C}[\partial]$-submodule of the conformal dual $L^{*c}$. We show that the contravariant functor ${ }^{0}$ is right adjoint to the contravariant functor ${ }^{*c}$. We define the Loc functor from the category of differential Lie coalgebras to the category of locally finite differential Lie coalgebras, associating to any differential Lie coalgebra $M$ the differential Lie coalgebra Loc$(M)$, defined as the largest locally finite differential Lie subcoalgebra of $M$. We prove that for any Lie conformal algebra $L$ that is free as a $\mathbb{C}[\partial]$-module, Loc$(L^{0})$ is the set of conformal linear maps on $L$ whose kernel contains an ideal of $L$ of cofinite rank. In general, $L^{0}$ will not be locally finite, so $\operatorname{Loc}\left(L^{0}\right) \varsubsetneqq L^{0}$. We present an example illustrating this.