Lie coalgebras

Abstract

AbstractA Lie coalgebra is a coalgebra whose comultiplication Δ : M → M ⊗ M satisfies the Lie conditions. Just as any algebra A whose multiplication ϕ : A ⊗ A → A is associative gives rise to an associated Lie algebra L(A), so any coalgebra C whose comultiplication Δ : C → C ⊗ C is associative gives rise to an associated Lie coalgebra Lc(C). The assignment C ↦ Lc(C) is functorial. A universal coenveloping coalgebra Uc(M) is defined for any Lie Lie coalgebra M by asking for a right adjoint Uc to Lc. This is analogous to defining a universal enveloping algebra U(L) for any Lie algebra L by asking for a left adjoint U to the functor L. In the case of Lie algebras, the unit (i.e., front adjunction) 1 → L o U of the adjoint functor pair U ⊣ L is always injective. This follows from the Poincaré-Birkhoff-Witt theorem, and is equivalent to it in characteristic zero (x = 0). It is, therefore, natural to inquire about the counit (i.e., back adjunction) Lc o Uc → 1 of the adjoint functor pair Lc ⊣ Uc.Theorem. For any Lie coalgebra M, the natural mapLc(UcM) → M is surjective if and only if M is locally finite, (i.e., each element of M lies in a finite dimensional sub Lie coalgebra of M).An example is given of a non locally finite Lie coalgebra. The existence of such an example is surprising since any coalgebra C whose diagonal Δ is associative is necessarily locally finite by a result of that theory. The present paper concludes with a development of an analog of the Poincaré-Birkhoff-Witt theorem for Lie algebras which we choose to call the Dual Poincaré-Birkhoff-Witt Theorem and abbreviate by “The Dual PBWθ.” The constraints of the present paper, however, allow only a sketch of this theorem. A complete proof will appear in a subsequent paper. The reader may, however, consult [12], in the meantime, for details. The Dual PBWθ shows for any locally finite Lie coalgebra M the existence (in χ = 0) of a natural isomorphism of the graded Hopf algebras 0E(UcM) and 0E(ScM) associated to UcM and to ScM = Uc(TrivM) when Uc(M) and Sc(M) are given the Lie filtrations. [Just as Uc(M) is the analog of the enveloping algebra U(L) of a Lie algebra L, so Sc(V) is the analog of the symmetric algebra S(V) on a vector space V. Triv(M) denotes the trivial Lie coalgebra structure on the underlying vector space of M obtained by taking the comultiplication to be the zero map.