Lie algebroids are curved Lie algebras

We show that there is an equivalence of ∞-categories between Lie algebroids and certain kinds of curved Lie algebras. For this we develop a method to study the ∞category of curved Lie algebras using the homotopy theory of algebras over a complete operad.

[1]  H. Busemann.  Advances in mathematics , 1961 .

[2]  Hsuan-Yi Liao,et al.  Formal exponential map for graded manifolds , 2015, 1508.02780.

[3]  V. Hinich.  Dwyer-Kan localization revisited , 2013, 1311.4128.

[4]  Vladimir Hinich.  Homological algebra of homotopy algebras , 1997 .

[5]  M. Kontsevich.  Deformation Quantization of Poisson Manifolds , 1997, q-alg/9709040.

[6]  B. Toën,et al.  Shifted Poisson structures and deformation quantization , 2015, 1506.03699.

[7]  K. Costello.  A geometric construction of the Witten genus, I , 2010, 1112.0816.

[8]  B. Vallette.  Homotopy theory of homotopy algebras , 2014, 1411.5533.

[9]  J. Maunder.  Koszul duality and homotopy theory of curved Lie algebras , 2015, 1512.01975.

[10]  P. Safronov,et al.  Shifted Symplectic Lie Algebroids , 2016, International Mathematics Research Notices.

[11]  A. Căldăraru,et al.  Curved A ∞ algebras and Landau – Ginzburg models , 2010 .

[12]  Y. Oh,et al.  Lagrangian Intersection Floer Theory: Anomaly and Obstruction, Part I , 2010 .

[13]  D. Gaitsgory,et al.  A study in derived algebraic geometry Volume II: Deformations, Lie theory and formal geometry , 2018 .

[14]  D. Calaque,et al.  Formal moduli problems and formal derived stacks , 2018, 1802.09556.

[15]  A. Lazarev,et al.  Cocommutative coalgebras: homotopy theory and Koszul duality , 2014, 1403.0774.

[16]  B. M. Fulk.  MATH , 1992 .

[17]  D. Calaque,et al.  On the Lie algebroid of a derived self-intersection , 2013, 1306.5260.

[18]  K. Hess.  Rational homotopy theory , 2011 .

[19]  S. Halperin.  Universal enveloping algebras and loop space homology , 1992 .

[20]  B. Fresse.  Modules over Operads and Functors , 2007, 0704.3090.

[21]  K. Mackenzie.  Lie groupoids and Lie algebroids in Differential Geometry: REFERENCES , 1987 .

[22]  G. Rinehart.  DIFFERENTIAL FORMS ON GENERAL COMMUTATIVE ALGEBRAS , 1963 .

[23]  V. Dotsenko,et al.  The twisting procedure , 2018, 1810.02941.

[24]  T. Nikolaus.  Rational and p-adic homotopy theory , 2016 .

[25]  L. V. Ovsiannikov.  The Lie Theory , 1982 .

[26]  Joris,et al.  Commentarii Mathematici Helvetici , 2008 .

[27]  L. Positselski.  Two kinds of derived categories, Koszul duality, and comodule-contramodule correspondence , 2009, 0905.2621.

[28]  Shilin Yu.  Dolbeault dga and L∞-algebroid of the formal neighborhood , 2017 .

[29]  D. joyce.  Algebraic Geometry over 𝐶^{∞}-rings , 2019, Memoirs of the American Mathematical Society.

[30]  E. Getzler.  Maurer-Cartan elements and homotopical perturbation theory , 2018, 1802.06736.

[31]  B. Hodson,et al.  The effect of passage in vitro and in vivo on the properties of murine fibrosarcomas. II. Sensitivity to cell-mediated cytotoxicity in vitro. , 1985, British Journal of Cancer.

[32]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[33]  R. Tennant.  Algebra , 1941, Nature.

[34]  Saulo Alves de Araujo,et al.  Identification of novel keloid biomarkers through Profiling of Tissue Biopsies versus Cell Cultures in Keloid Margin specimens Compared to adjacent Normal Skin , 2010, Eplasty.

[35]  J. Pridham.  Unifying derived deformation theories , 2007, 0705.0344.

[36]  Adv , 2019, International Journal of Pediatrics and Adolescent Medicine.

[37]  Eduardo A. Fierro.  Structures , 2003, Composite Architecture.

[38]  J. Maunder.  Unbased rational homotopy theory:a Lie algebra approach , 2015, 1511.07669.

[39]  Joost Nuiten,et al.  Homotopical Algebra for Lie Algebroids , 2017, Applied Categorical Structures.

[40]  G. Vezzosi.  A model structure on relative dg-Lie algebroids , 2013, 1304.6049.

[41]  J. Nuiten.  Koszul duality for Lie algebroids , 2017, Advances in Mathematics.

[42]  G. Felder,et al.  Relative formality theorem and quantisation of coisotropic submanifolds , 2005, math/0501540.

[43]  M. Bergh,et al.  Hochschild (Co)homology for Lie Algebroids , 2009, 0908.2630.

[44]  K. Fukaya.  Deformation theory, homological algebra and mirror symmetry , 2001 .

[45]  Ieke Moerdijk,et al.  Axiomatic homotopy theory for operads , 2002, math/0206094.

[46]  Joseph Hirsh,et al.  Curved Koszul duality theory , 2010, 1008.5368.

[47]  Christopher L. Rogers,et al.  A version of the Goldman–Millson theorem for filtered L∞-algebras , 2015 .

[48]  S. Ana,et al.  Topology , 2018, International Journal of Mathematics Trends and Technology.

[49]  J. Stasheff,et al.  Deformation theory and rational homotopy type , 2012, 1211.1647.

[50]  P. Steerenberg,et al.  Targeting pathophysiological rhythms: prednisone chronotherapy shows sustained efficacy in rheumatoid arthritis. , 2010, Annals of the rheumatic diseases.

[51]  Christopher L. Rogers.  Complete filtered L∞-algebras and their homotopy theory , 2020 .

[52]  Y. Oh,et al.  Lagrangian intersection floer theory : anomaly and obstruction , 2009 .

[53]  Sarah Whitehouse,et al.  Model category structures and spectral sequences , 2018, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.