The goal of the article under review is to transfer the differential graded Lie algebra (dgla) structure from a dgla \({\mathfrak g}\) to a deformation retract \(M\). The transferred structure will be a weaker structure, namely the structure of an \(L_{\infty}\)-algebra. This transfer result is called the Lie Algebra Perturbation Lemma. The present article gives full proofs of and extends (some of) the results of [J. Huebschmann and J. Stasheff, “Formal solution of the master equation via HPT and deformation theory”, Forum Math. 14, No. 6, 847–868 (2002; Zbl 1036.17016)] to possibly non-zero differential on \(M\). The present article will be extended to general \(L_{\infty}\)-algebras \({\mathfrak g}\) instead of dglas in [J. Huebschmann, “The sh-Lie algebra perturbation lemma”, Forum Math. 23, No. 4, 669–691 (2011; Zbl 1243.17012)].
Let us give a more detailed description of the main result. All differentials will be of degree \(-1\) (homological convention). The commutative ground ring \(R\) is supposed to contain the rational numbers. Let \({\mathfrak g}\) be a dgla and \[ (M\rightleftarrows{\mathfrak g},h) \] be a contraction of chain complexes, i.e.,

1.

\(M\) is a chain complex,

2.

there are chain maps \(\pi:M\to{\mathfrak g}\) and \(\nabla:{\mathfrak g}\to M\),

3.

there is a morphism \(h:M\to M\) of the underlying graded modules of degree \(1\)

such that

1.

\(\pi\circ\nabla=\text{id}_M\),

2.

\(d\circ h+h\circ d=\text{id}_{\mathfrak g}-\nabla\circ\pi\),

3.

\(\pi\circ h=0\),\(h\circ\nabla=0\),\(h\circ h=0\).

The first two conditions state that while \(\pi\circ\nabla\) is the identity, \(\nabla\circ\pi\) is only homotopic to the identity. The conditions of the last line will be called side conditions.
An \(L_{\infty}\)-algebra structure on a graded space \(M\) can be most compactly defined as a codifferential on the cofree symmetric coalgebra \({\mathcal S}^c:={\mathcal S}^c[sM]\) on the suspended graded space \(sM\). By cofreeness, morphisms \({\mathcal S}^c\to{\mathcal S}^c\) may be defined by specifying only a map \({\mathcal S}^c\to M\). As we start from a chain complex \(M\), \({\mathcal S}^c\) carries already a coalgebra differential \(d\), thus the \(L_{\infty}\)-algebra structure to be defined will be a coalgebra perturbation of this differential, i.e., a degree \(-1\) map \({\mathcal D}:{\mathcal S}^c\to{\mathcal S}^c\) such that \(d+{\mathcal D}\) is compatible with the coalgebra structure and \((d+{\mathcal D})^2=0\). Equipped with the new coalgebra differential \(d+{\mathcal D}\), \({\mathcal S}^c\) will be denoted by \({\mathcal S}^c_{\mathcal D}\).
The main theorem of the article under review asserts now that given a dgla \({\mathfrak g}\), a chain complex \(M\) (with differential \(d\)) and a contraction \((M\rightleftarrows{\mathfrak g},h)\), there exists a coalgebra perturbation \({\mathcal D}\) (of the induced differential \(d\)) on \({\mathcal S}^c\) such that the construction induces a contraction \[ ({\mathcal S}^c_{\mathcal D}\rightleftarrows{\mathcal S}^c [s{\mathfrak g}],H), \] where \({\mathcal S}^c[s{\mathfrak g}]\) is the standard (Chevalley-Eilenberg-) coalgebra on \({\mathfrak g}\). All constructions are natural in the dgla structure on \({\mathfrak g}\) and the contraction \((M\rightleftarrows{\mathfrak g},h)\).
The operator \({\mathcal D}\) is constructed by induction, together with a twisting cochain \(\tau:{\mathcal S}^c_{\mathcal D}\to{\mathfrak g}\), as follows:

1.

\(\tau^1=\nabla\circ\tau_M:{\mathcal S}^c\to{\mathfrak g}\),

2.

\(\tau^j=\frac{1}{2}h([\tau^1,\tau^{j-1}]+\dots+[\tau^{j-1},\tau^1]) :{\mathcal S}^c\to{\mathfrak g}\), \(j\geq 2\),

3.

\(\tau=\tau^1+\tau^2+\dots:{\mathcal S}^c\to{\mathfrak g}\),

4.

\({\mathcal D}={\mathcal D}^1+{\mathcal D}^2+\dots:{\mathcal S}^c \to {\mathcal S}^c\),

where, for \(j\geq 1\), the operator \({\mathcal D}^j\) is the coderivation of \({\mathcal S}^c[sM]\) determined by \[ \tau_M\circ{\mathcal D}^j=\frac{1}{2}\pi([\tau^1,\tau^{j}]+\dots+ [\tau^{j},\tau^1]):{\mathcal S}^c_{j+1}\to M. \]
Here \(\tau_M:{\mathcal S}^c[sM]\to M\) is the composition of the canonical projection onto \(sM\) with the desuspension map \(sM\to M\). The constructed sums \(\tau\) and \({\mathcal D}\) are in general infinite. They are constructed by induction using the filtration on \({\mathcal S}^c[sM]\). As \({\mathcal S}^c[sM]\) is cocomplete, a specific element lies in a subspace of finite filtration, and on such an element, only a finite number of terms in the sum will be non-zero. The above side conditions are necessary to initiate the induction procedure.
The structure of the result and its proof are explained in Section 2. Section 3 contains a detailed discussion justifying that in case the commutative ground ring \(R\) contains the rational numbers, the cofree coalgebra on any chain complex over \(R\) exists. Section 4 contains the inductive construction of \(\tau\) and \({\mathcal D}\). Section 5 contains the construction of the contraction \(({\mathcal S}^c_{\mathcal D}\rightleftarrows{\mathcal S}^c [s{\mathfrak g}],H)\) and therefore concludes the proof of the main theorem.
For the entire collection see [Zbl 1203.00021].