[Paper Review] A relative version of the ordinary perturbation lemma

Marco Manetti|arXiv (Cornell University)|Feb 3, 2010

Algebraic structures and combinatorial models12 references3 citations

TL;DR

This paper introduces a relative version of the ordinary perturbation lemma for chain complexes, enabling a direct and elementary proof of the homotopy transfer theorem for $L_{ u}$-algebras. By adapting the perturbation lemma to work relative to a differential graded subspace with specific properties, the author bypasses the failure of the tensor trick in symmetric algebras and establishes functorial formulas for transferring $L_{ u}$-structures via recursive formulas on rooted trees.

ABSTRACT

The perturbation lemma and the homotopy transfer for L-infinity algebras is proved in a elementary way by using a relative version of the ordinary perturbation lemma for chain complexes and the coalgebra perturbation lemma.

Motivation & Objective

• To overcome the failure of the tensor trick in symmetric algebras, which obstructs standard homotopy transfer for $L_{ u}$-algebras.
• To provide a direct, elementary proof of the homotopy transfer theorem for $L_{ u}$-algebras without relying on advanced homological perturbation theory.
• To establish a relative version of the perturbation lemma that applies when a new differential $ ilde{d}$ preserves a subspace $A \subset N$ with suitable properties.
• To show that the inclusion map $\imath_{\partial}$ into the symmetric coalgebra $\overline{S}(V)$ satisfies a recursive formula involving projections and perturbations.
• To demonstrate that the resulting transfer map is a morphism of differential graded coalgebras via combinatorial tree expansions.

Proposed method

• Introduce a relative perturbation lemma where the perturbation $\partial$ preserves a differential graded subspace $A \subset N$, ensuring compatibility with the homotopy structure.
• Use the standard perturbation lemma in conjunction with the coalgebra perturbation lemma to derive explicit formulas for the transferred differential on $M$.
• Define the transferred inclusion $\imath_{\partial}$ recursively via the equation $p\imath_{\partial} = p\imath + kp\partial\imath_{\partial}$, where $p$ is the projection to $V$.
• Leverage the fact that $\pi_{\partial}\imath_{\partial} = \text{id}$ on $\overline{S}(W)$, ensuring the transfer is well-defined and invertible up to homotopy.
• Show that the map $\imath_{\partial}$ is the unique morphism of symmetric graded coalgebras satisfying the recursive formula, with a combinatorial interpretation in terms of rooted trees.
• Establish that $\imath_{\partial}\colon (\overline{S}(W), d + \pi\partial\imath_{\partial}) \to (\overline{S}(V), d + \partial)$ is a morphism of differential graded coalgebras.

Experimental results

Research questions

• RQ1 Can the homotopy transfer for $L_{\infty}$-algebras be proven without relying on the tensor trick, which fails under symmetrization?
• RQ2 Is there a relative version of the perturbation lemma that applies when the perturbation preserves a subspace with controlled homotopy properties?
• RQ3 Can the transferred $L_{\infty}$-structure be described via a recursive formula on rooted trees in the symmetric coalgebra setting?
• RQ4 Does the inclusion map $\imath_{\partial}$ into the symmetric coalgebra $\overline{S}(V)$ preserve the differential graded coalgebra structure under perturbation?
• RQ5 Can the homotopy classification of $L_{\infty}$-algebras be used to guarantee the existence of a compatible morphism $\Pi\colon \overline{S}(V) \to \overline{S}(W)$ with $\Pi\imath_{\partial} = \text{id}$?

Key findings

• The relative perturbation lemma provides a direct and elementary proof of the homotopy transfer theorem for $L_{\infty}$-algebras, avoiding the need for nontrivial additional work in the symmetric setting.
• The transferred inclusion $\imath_{\partial}\colon \overline{S}(W) \to \overline{S}(V)$ satisfies the recursive formula $p\imath_{\partial} = p\imath + kp\partial\imath_{\partial}$, which uniquely determines it as a morphism of symmetric graded coalgebras.
• The map $\imath_{\partial}$ is shown to be a morphism of differential graded coalgebras, ensuring compatibility with the $L_{\infty}$-structure.
• The recursive formula for $\imath_{\partial}$ admits a combinatorial interpretation as a sum over rooted trees, providing an explicit construction of the transfer.
• The existence of a morphism $\Pi\colon \overline{S}(V) \to \overline{S}(W)$ with $\Pi\imath_{\partial} = \text{id}$ is guaranteed by the homotopy classification of $L_{\infty}$-algebras, ensuring the transfer is well-behaved.
• The proof establishes that $\pi_{\partial}\imath_{\partial} = \text{id}$ on $\overline{S}(W)$, confirming the transfer is a quasi-inverse up to homotopy.