The Lie algebra perturbation lemma. (English) Zbl 1243.17011
Cattaneo, Alberto S. (ed.) et al., Higher structures in geometry and physics. In honor of Murray Gerstenhaber and Jim Stasheff. Basel: Birkhäuser (ISBN 978-0-8176-4734-6/hbk; 978-0-8176-4735-3/ebook). Progress in Mathematics 287, 159-179 (2011).
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.,
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:
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].
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 \(1\)
- 1.
- \(\pi\circ\nabla=\text{id}_M\),
- 2.
- \(d\circ h+h\circ d=\text{id}_{\mathfrak g}-\nabla\circ\pi\),
- 3.
- \(\pi\circ h=0\),\(h\circ\nabla=0\),\(h\circ h=0\).
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.
- \({\mathcal D}={\mathcal D}^1+{\mathcal D}^2+\dots:{\mathcal S}^c \to {\mathcal S}^c\),
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].
Reviewer: Friedrich Wagemann (Nantes)
MSC:
17B55 | Homological methods in Lie (super)algebras |
17B70 | Graded Lie (super)algebras |
55P62 | Rational homotopy theory |
16E45 | Differential graded algebras and applications (associative algebraic aspects) |
16T15 | Coalgebras and comodules; corings |
18G35 | Chain complexes (category-theoretic aspects), dg categories |