Differential Lie Coalgebras and Lie Conformal Algebras

TL;DR

This paper introduces a functorial relationship between Lie conformal algebras and differential Lie coalgebras, exploring their properties, adjunctions, and local finiteness, with applications to conformal linear maps.

Contribution

It defines a functor from Lie conformal algebras to differential Lie coalgebras and studies the Loc functor, establishing adjunctions and properties for free modules.

Findings

The functor from Lie conformal algebras to differential Lie coalgebras is constructed.

The Loc functor identifies the largest locally finite differential Lie subcoalgebra.

For free modules, Loc(L^0) corresponds to conformal maps with cofinite ideal kernels.

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.