Lie-Rinehart algebras \(\simeq\) acyclic Lie \(\infty \)-algebroids. (English) Zbl 1482.18016
The article under review generalizes to Lie-Rinehart algebras, earlier work by the first author with S. Lavau and T. Strobl [C. Laurent-Gengoux et al., Doc. Math. 25, 1571–1652 (2020; Zbl 1453.53033)], where the homotopy class of an \(L_{\infty}\)-algebroid was assigned to a singular foliation. In fact, the current paper improves this result, in the sense that the length of the \(L_{\infty}\)-algebroid is allowed to be infinite. Moreover, the construction given in this work provides a complex which allows the computation of the Tor functor.
Reviewer: Iakovos Androulidakis (Athína)
MSC:
| 18N99 | Higher categories and homotopical algebra |
| 53C12 | Foliations (differential geometric aspects) |
| 55U35 | Abstract and axiomatic homotopy theory in algebraic topology |
| 16S38 | Rings arising from noncommutative algebraic geometry |
| 16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |
| 17B60 | Lie (super)algebras associated with other structures (associative, Jordan, etc.) |
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
Keywords:
Lie-Rinehart algebras; Lie algebroids; algebras up to homotopy; Lie infinity algebras; singular foliations; algebraic geometryCitations:
Zbl 1453.53033References:
| [1] | Androulidakis, Iakovos; Zambon, Marco, Smoothness of holonomy covers for singular foliations and essential isotropy, Math. Z., 275, 921-951 (2013) · Zbl 1292.57020 |
| [2] | Androulidakis, Iakovos; Zambon, Marco, Integration of singular algebroids (2018), arXiv · Zbl 1292.57020 |
| [3] | Bonavolontà, Giuseppe; Poncin, Norbert, On the category of Lie n-algebroids, J. Geom. Phys., 73, 70-90 (2013) · Zbl 1332.58005 |
| [4] | Campos, Ricardo, BV formality, Adv. Math., 306, 807-851 (2017) · Zbl 1386.18034 |
| [5] | Campos, Ricardo, Homotopy equivalence of shifted cotangent bundles, J. Lie Theory, 29, 3, 629-646 (2019) · Zbl 1442.58003 |
| [6] | Cattaneo, Alberto S.; Felder, Giovanni, Coisotropic submanifolds in Poisson geometry and branes in the Poisson sigma model, Lett. Math. Phys., 69, 157-175 (2004) · Zbl 1065.53063 |
| [7] | Cerveau, Dominique, Distributions involutives singulières, Ann. Inst. Fourier (Grenoble), 29, 3 (1979), xii, 261-294 · Zbl 0419.58002 |
| [8] | Chen, Zhuo; Stiénon, Mathieu; Xu, Ping, From Atiyah classes to homotopy Leibniz algebras, Commun. Math. Phys., 341, 309-349 (2016) · Zbl 1395.53090 |
| [9] | Dazord, Pierre, Feuilletages à singularités, Indag. Math., 47, 21-39 (1985) · Zbl 0584.57016 |
| [10] | Debord, Claire, Holonomy groupoids of singular foliations, J. Differ. Geom., 58, 467-500 (07 2001) · Zbl 1034.58017 |
| [11] | Eisenbud, David, Commutative Algebra. With a View Toward Algebraic Geometry, vol. 150 (1995), Springer-Verlag: Springer-Verlag Berlin · Zbl 0819.13001 |
| [12] | Frégier, Yaël; Juarez-Ojeda, Rigel A., Homotopy theory of singular foliations (2018) |
| [13] | Frölicher, Alfred; Nijenhuis, Albert, Theory of vector-valued differential forms. I: Derivations in the graded ring of differential forms, Ned. Akad. Wet., Proc., Ser. A, 59, 338-350, 351-359 (1956) · Zbl 0079.37502 |
| [14] | Andreas Gathmann, Commutative Algebra, Class Notes TU Kaiserslautern 2013/14. |
| [15] | Hartshorne, Robin, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52 (1977), Springer: Springer New York, NY · Zbl 0367.14001 |
| [16] | Huebschmann, Johannes, Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras, Ann. Inst. Fourier, 48, 425-440 (1998) · Zbl 0973.17027 |
| [17] | Huebschmann, Johannes, Lie-Rinehart Algebras, Descent, and Quantization, Fields Inst. Commun., vol. 43 (2004), Amer. Math. Soc.: Amer. Math. Soc. Providence, RI · Zbl 1096.17006 |
| [18] | Kapranov, Maxim, Rozansky-Witten invariants via Atiyah classes, Compos. Math., 115, 71-113 (1999) · Zbl 0993.53026 |
| [19] | Kassel, Christian, Quantum Groups, Graduate Texts in Mathematics, vol. 155 (2012), Springer · Zbl 0808.17003 |
| [20] | Kjeseth, Lars, A homotopy Lie-Rinehart resolution and classical BRST cohomology, Homol. Homotopy Appl., 3, 165-192 (2001) · Zbl 1054.17016 |
| [21] | Kjeseth, Lars, Homotopy Rinehart cohomology of homotopy Lie-Rinehart pairs, Homol. Homotopy Appl., 3, 139-163 (2001) · Zbl 1005.17016 |
| [22] | Kolář, Ivan; Michor, Peter W.; Slovák, Jan, Natural Operations in Differential Geometry (1993), Springer-Verlag: Springer-Verlag Berlin · Zbl 0782.53013 |
| [23] | Kontsevich, Maxim, Deformation quantization of Poisson manifolds, Lett. Math. Phys., 66, 157-216 (2003) · Zbl 1058.53065 |
| [24] | Kosmann-Schwarzbach, Yvette, Derived brackets, Lett. Math. Phys., 69, 61-87 (2004) · Zbl 1055.17016 |
| [25] | Lada, Tom; Stasheff, Jim, Introduction to SH Lie algebras for physicists, Int. J. Theor. Phys., 32, 1087-1103 (1993) · Zbl 0824.17024 |
| [26] | Laurent-Gengoux, Camille; Lavau, Sylvain; Strobl, Thomas, The universal Lie ∞-algebroid of a singular foliation, Doc. Math., 25, 1571-1652 (2020) · Zbl 1453.53033 |
| [27] | Laurent-Gengoux, Camille; Stiénon, Mathieu; Xu, Ping, Poincaré-Birkhoff-Witt isomorphisms and Kapranov dg-manifolds, Adv. Math., 387 (2021) · Zbl 1468.58004 |
| [28] | Lavau, Sylvain, Lie ∞-algebroids and singular foliations (2017), Ph-D |
| [29] | Mackenzie, Kirill C. H., General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, vol. 213 (2005), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1078.58011 |
| [30] | Matsumura, Hideyuki, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (1989), Cambridge University Press: Cambridge University Press Cambridge, translated by M. Reid · Zbl 0666.13002 |
| [31] | May, Jon Peter, Cohen-Macauley and regular local rings, On-line 06-2021 |
| [32] | Sati, Hisham; Schreiber, Urs; Stasheff, Jim, Twisted differential string and fivebrane structures, Commun. Math. Phys., 315, 1, 169-213 (2012) · Zbl 1252.81110 |
| [33] | Ševera, Pavol, Some title containing the words “homotopy” and “symplectic”, e.g. this one (2001) · Zbl 1102.58010 |
| [34] | Ševera, Pavol; Širaň, Michal, Integration of differential graded manifolds, Int. Math. Res. Not., 20, 6769-6814 (2020) · Zbl 1468.58002 |
| [35] | Tseng, Hsian-Hua; Zhu, Chenchang, Integrating Lie algebroids via stacks, Compos. Math., 142, 251-270 (2006) · Zbl 1111.58019 |
| [36] | Villatoro, Joel, On sheaves of Lie-Rinehart algebras (2020) |
| [37] | Vitagliano, Luca, On the strong homotopy associative algebra of a foliation, Commun. Contemp. Math., 17 (2015) · Zbl 1316.53025 |
| [38] | Vitagliano, Luca, Representations of homotopy Lie-Rinehart algebras, Math. Proc. Camb. Philos. Soc., 158, 155-191 (2015) · Zbl 1371.17018 |
| [39] | Voronov, Theodore, Higher derived brackets and homotopy algebras, J. Pure Appl. Algebra, 202, 133-153 (2005) · Zbl 1086.17012 |
| [40] | Voronov, Theodore, Q-manifolds and higher analogs of Lie algebroids, (XXIX Workshop on Geometric Methods in Physics. XXIX Workshop on Geometric Methods in Physics, AIP Conf. Proc., vol. 1307 (2010)), 191-202 · Zbl 1260.53133 |
| [41] | Weibel, Charles, An Introduction to Homological Algebra, Cambridge Studies in Advanced Mathematics (2014), Cambridge University Press · Zbl 0797.18001 |
| [42] | Zambon, Marco, Holonomy transformations for Lie subalgebroids (2018) |
| [43] | Zambon, Marco; Androulidakis, Iakovos, Singular subalgebroids (2018) · Zbl 1352.53021 |
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