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**Noncommutative formality implies commutative and Lie formality.**
*(English)*
Zbl 1410.55006

When dealing with associative, commutative or Lie differential graded algebras (a.k.a. dgas), formality is an important and much studied concept. For example, an associative dga \(A\) is formal if there is a chain of quasi-isomorphisms of associative dgas relating \(A\) to its cohomology algebra \(H^*A\), viewed as a dga equipped with the trivial differential. In the paper under review, the author studies how different notions of formality imply each other. One main result states that if \(A\) is a commutative dga over a field of characteristic 0, then \(A\) is formal as a commutative dga if and only if \(A\) is formal as an associative dga. Another main result states that a differential graded Lie algebra over a field of characteristic 0 is formal if and only if its universal enveloping algebra is formal as an associative dga. These two results have applications to rational homotopy theory where one is interested in the formality of cochain algebras over the rationals.

One important tool for the proofs of the main results in this paper is a theorem by J.-L. Loday and B. Vallette [Algebraic operads. Berlin: Springer (2012; Zbl 1260.18001)] that characterizes formality over a Koszul operad \(\mathcal{P}\) in terms of a property of algebras over the cobar construction \(\mathcal{P}_\infty\) of the Koszul dual operad of \(\mathcal{P}\). For \(\mathcal{P}\) being the associative, the commutative, or the Lie operad, the author then shows that formality of minimal \(\mathcal{P}_\infty\) algebras is detected by obstruction classes in an operadic chain complex. This, for example, specializes to results by T. V. Kadeishvili [in: Tr. Tbilis. Mat. Inst. Razmadze 91, 19–27 (1988; Zbl 0717.55011)] about minimal associative dgas and the Hochschild complex. The non-trivial implications of the above main results are then established by showing that suitable maps between obstruction groups are injective so that the vanishing of the image of obstruction classes in the target implies the vanishing of the classes in the source.

One important tool for the proofs of the main results in this paper is a theorem by J.-L. Loday and B. Vallette [Algebraic operads. Berlin: Springer (2012; Zbl 1260.18001)] that characterizes formality over a Koszul operad \(\mathcal{P}\) in terms of a property of algebras over the cobar construction \(\mathcal{P}_\infty\) of the Koszul dual operad of \(\mathcal{P}\). For \(\mathcal{P}\) being the associative, the commutative, or the Lie operad, the author then shows that formality of minimal \(\mathcal{P}_\infty\) algebras is detected by obstruction classes in an operadic chain complex. This, for example, specializes to results by T. V. Kadeishvili [in: Tr. Tbilis. Mat. Inst. Razmadze 91, 19–27 (1988; Zbl 0717.55011)] about minimal associative dgas and the Hochschild complex. The non-trivial implications of the above main results are then established by showing that suitable maps between obstruction groups are injective so that the vanishing of the image of obstruction classes in the target implies the vanishing of the classes in the source.

Reviewer: Steffen Sagave (Nijmegen) (MR3686405)

### MSC:

55P62 | Rational homotopy theory |

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\textit{B. Saleh}, Algebr. Geom. Topol. 17, No. 4, 2523--2542 (2017; Zbl 1410.55006)

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