Differential Lie Coalgebras and Lie Conformal Algebras

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 L0, defined as the maximal good C[∂]-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 C[∂]-module, Loc(L0) is the set of conformal linear maps on L whose kernel contains an ideal of L of cofinite rank. In general, L0 will not be locally finite, so Loc L0 L0. We present an example illustrating this.