[Paper Review] The sh-Lie algebra perturbation Lemma

Johannes Huebschmann|arXiv (Cornell University)|Oct 10, 2007

Advanced Topics in Algebra4 citations

TL;DR

This paper establishes a perturbation lemma for sh-Lie (L∞) algebras, extending the classical perturbation lemma for chain complexes to the setting of homotopy Lie algebras. Given a contraction of chain complexes over a ring containing ℚ, an sh-Lie algebra structure on the original complex induces a natural sh-Lie structure on the smaller complex, with an associated sh-equivalence implemented via a Lie algebra twisting cochain and its adjoint, preserving algebraic structure under homotopy.

ABSTRACT

Let R be a commutative ring which contains the rationals as a subring and let g be a chain complex. Suppose given an sh-Lie algebra structure on g, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra T' on the suspension of g and write the perturbed coalgebra as T". Suppose, furthermore, given a contraction of g onto a chain complex M. We show that the data determine an sh-Lie algebra structure on M, that is, a coalgebra perturbation of the coalgebra differential on the cofree coaugmented differential graded cocommutative coalgebra S' on the suspension of M, a Lie algebra twisting cochain from the perturbed coalgebra S" to the loop Lie algebra L on the perturbed coalgebra T", and an extension of this Lie algebra twisting cochain to a contraction of chain complexes from the Cartan-Chevalley-Eilenberg coalgebra on L onto S" which is natural in the data. For the special case where M and g are connected we also construct an explicit extension of the perturbed retraction to an sh-Lie map. This approach includes a very general solution of the master equation.

Motivation & Objective

• To generalize the classical perturbation lemma for chain complexes to the context of sh-Lie algebras (L∞-algebras), where higher homotopies arise from failure of strict algebraic identities.
• To address the subtlety that homotopies of morphisms between cocommutative coalgebras do not generally preserve sh-Lie algebra structure, requiring a refined framework.
• To construct a natural sh-equivalence between a chain complex M and a larger complex 𝔤 equipped with an sh-Lie algebra structure, using the given contraction and coalgebra perturbations.
• To provide an explicit construction of an sh-inverse for the retraction map under connectivity assumptions, ensuring the equivalence is computable and structure-preserving.

Proposed method

• Utilizes the symmetric coalgebra functor 𝒮ᶜ, suspension operator s, loop Lie algebra functor ℒ, and classifying coalgebra functor 𝒞 to translate algebraic structures between complexes.
• Defines an sh-Lie algebra structure on M via a coalgebra perturbation 𝒟 of the differential on 𝒮ᶜ[sM], induced from the original perturbation ∂ on 𝔤.
• Constructs a Lie algebra twisting cochain τ: 𝒮ᶜ𝒟[sM] → ℒ𝒮ᶜ∂[s𝔤], which encodes the algebraic equivalence between M and 𝔤 at the homotopy level.
• Establishes a contraction of chain complexes between 𝒮ᶜ𝒟[sM] and 𝒞[ℒ𝒮ᶜ∂[s𝔤]], with the injection 𝒯: 𝒮ᶜ𝒟[sM] → 𝒞[ℒ𝒮ᶜ∂[s𝔤]] being a morphism of coaugmented dg coalgebras.
• Uses the adjoint of the universal twisting cochain to relate the classifying coalgebra of the loop algebra to the symmetric coalgebra, ensuring compatibility with sh-structures.
• Applies Lemma 4.2 to construct an explicit homotopy hC between the identity and the composite of twisting cochains when M and 𝔤 are connected, yielding an explicit sh-inverse.

Experimental results

Research questions

• RQ1 How can the classical perturbation lemma for chain complexes be extended to preserve sh-Lie algebra structures under homotopy?
• RQ2 What conditions ensure that a contraction of chain complexes induces a well-defined sh-Lie algebra structure on the smaller complex?
• RQ3 Can a natural sh-equivalence be constructed between two sh-Lie algebras related by a contraction, and if so, how is it encoded algebraically?
• RQ4 Under what conditions can the retraction map in a contraction be promoted to an explicit sh-isomorphism via higher homotopies?
• RQ5 Why does the naive notion of homotopy fail to preserve sh-Lie algebra structure, and how can this be resolved using higher homotopy constructions?

Key findings

• An sh-Lie algebra structure on 𝔤 induces a natural sh-Lie algebra structure on M via a coalgebra perturbation 𝒟 of the differential on 𝒮ᶜ[sM].
• The data of the contraction and the sh-Lie structure on 𝔤 determine a Lie algebra twisting cochain τ: 𝒮ᶜ𝒟[sM] → ℒ𝒮ᶜ∂[s𝔤], encoding the homotopy equivalence.
• The adjoint of τ yields a morphism 𝒯: 𝒮ᶜ𝒟[sM] → 𝒞[ℒ𝒮ᶜ∂[s𝔤]] that is a morphism of coaugmented dg coalgebras, establishing a contraction of chain complexes.
• This contraction is natural in the data and realizes an sh-equivalence between (M, 𝒟) and (𝔤, ∂), with the equivalence preserved under the algebraic structure.
• For connected M and 𝔤, an explicit sh-inverse to the retraction is constructed via a homotopy hC between twisting cochains, ensuring the equivalence is computable.
• The construction shows that while naive homotopies leave the world of sh-Lie algebras, higher homotopy constructions (e.g., via BΩ𝒮ᶜ) keep the structure within the category of sh-objects.