Question is as mentioned in the title:


Are there any introductory notes on deformation theory that are easier to read for differential geometers?


I am learning about differential graded Lie algebras (and $L\_\infty$-algebras). I am aware of Marco Manetti's notes [Deformation theory via differential graded Lie algebras](https://arxiv.org/abs/math/0507284).


Are there any other notes that are easier to read for people who are trained in differential geometry?


I know some commutative algebra, for example definition (and few properties) of Noetherian, Artinian rings. I can recall (or read) some more commutative algebra if required.


I read from the paper [From Lie Theory to Deformation Theory and Quantization](https://arxiv.org/abs/0704.2213) that



> 
> "Deformation Theory is a natural generalization of Lie Theory, from Lie groups and their linearization, Lie algebras, to differential graded Lie algebras and their higher order deformations, quantum groups."
> 
> 
> 


So, this gave some hope that there would be some notes from the point of view of Lie theory/differential geometry.



(An introduction to) deformation theory (written) for differential geometers
A classical subject in deformation theory with a differential-geometric flavour is that of deformation quantization. There are several good introductory references I can recommend, all of which introduce the Maurer–Cartan formalism for DG Lie algebras in deformation theory:


* *Cattaneo, Alberto S.*, Formality and star products (lecture notes taken by D. Indelicato), Gutt, Simone (ed.) et al., Poisson geometry, deformation quantisation and group representations. Cambridge: Cambridge University Press (ISBN 0-521-61505-4/pbk). London Mathematical Society Lecture Note Series 323, 79-144 (2005). [ZBL1077.53074](https://zbmath.org/?q=an:1077.53074).
* [some later chapters in French] *Cattaneo, Alberto; Keller, Bernhard; Torossian, Charles; Bruguières, Alain*, Déformation, quantification, théorie de Lie, Panoramas et Synthèses 20. Paris: Société Mathématique de France (ISBN 2-85629-183-X/pbk). vii, 186 p. (2005). [ZBL1093.53095](https://zbmath.org/?q=an:1093.53095).
* [in German] *Waldmann, Stefan*, [**Poisson-Geometrie und Deformationsquantisierung**](https://doi.org/10.1007/978-3-540-72518-3), Berlin: Springer (ISBN 978-3-540-72517-6/pbk). xii, 612 p. (2007). [ZBL1139.53001](https://zbmath.org/?q=an:1139.53001).
* *Esposito, Chiara*, [**Formality theory. From Poisson structures to deformation quantization**](https://doi.org/10.1007/978-3-319-09290-4), SpringerBriefs in Mathematical Physics 2. Cham: Springer (ISBN 978-3-319-09289-8/pbk; 978-3-319-09290-4/ebook). xii, 90 p. (2015). [ZBL1301.81003](https://zbmath.org/?q=an:1301.81003).


Another reference for deformation quantization, taking a Lie-theoretic point of view (but whose prerequisites are only "[basic linear algebra and differential geometry](https://ems.press/books/elm/93)"), is


* *Calaque, Damien; Rossi, Carlo A.*, [**Lectures on Duflo isomorphisms in Lie algebra and complex geometry**](https://doi.org/10.4171/096), EMS Series of Lectures in Mathematics. Zürich: European Mathematical Society (EMS) (ISBN 978-3-03719-096-8/pbk). viii, 106 p. (2011). [ZBL1220.53006](https://zbmath.org/?q=an:1220.53006).


Besides Manetti's new book


* *Manetti, Marco*, [**Lie methods in deformation theory**](https://doi.org/10.1007/978-981-19-1185-9), Springer Monographs in Mathematics. Singapore: Springer (ISBN 978-981-19-1184-2). xii, 574 p. (2022). [ZBL07529473](https://zbmath.org/?q=an:07529473).


which centers around deformations of compact complex manifolds, there is also a freely available shorter "book" (a 183-page article) by Manetti


* *Manetti, Marco*, [**Lectures on deformations of complex manifolds. Deformations from differential graded viewpoint**](https://arxiv.org/abs/math/0507286), Rend. Mat. Appl., VII. Ser. 24, No. 1, 1-183 (2004). [ZBL1066.58010](https://zbmath.org/?q=an:1066.58010) [PDF](https://www1.mat.uniroma1.it/ricerca/rendiconti/visualizza-volume.cgi?anno=2004&volume=24&issue=1).


where you can also find differential forms in the main examples.


For the passage from the DG Lie algebra / L$\_\infty$ algebra viewpoint to derived deformation theory see [this question](https://mathoverflow.net/q/392885/163420).


