Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 074, 33 pages      arXiv:2412.12935      https://doi.org/10.3842/SIGMA.2026.074

$\mathcal{L}$-Lie Algebroids over Topological Ringed Spaces

Mainak Poddar a and Abhishek Sarkar ab
a) Department of Mathematics, Indian Institute of Science Education and Research Pune, India
b) School of Computing, MIT Art, Design and Technology University, Loni Kalbhor, Pune, India

Received October 14, 2025, in final form August 01, 2026; Published online August 11, 2026

Abstract
Lie algebroids over a topological ringed space provides a unified framework to study various geometric structures. This geometric concept is intimately connected with well-known algebraic structures, including Gerstenhaber algebras and Batalin-Vilkovisky algebras. We introduce more general concepts such as $\mathcal{L}$-Lie algebroids and $\mathcal{A}$-Gerstenhaber algebras, associated with a given Lie algebroid $\mathcal{L}$ and Gerstenhaber algebra $\mathcal{A}$ over a topological ringed space, respectively. Following this, we explore how several standard correspondences extend within this broader framework.

Key words: Lie algebroids; tangent sheaf; Gerstenhaber algebras; BV-algebras; universal enveloping algebra; Lie algebroid connections; Lie algebroid (co)homology.

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References

  1. Androulidakis I., Skandalis G., The holonomy groupoid of a singular foliation, J. Reine Angew. Math. 626 (2009), 1-37, arXiv:math.OA/0612370.
  2. Atiyah M.F., Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181-207.
  3. Biswas I., Dey A., Poddar M., Equivariant principal bundles and logarithmic connections on toric varieties, Pacific J. Math. 280 (2016), 315-325, arXiv:1507.02415.
  4. Biswas I., Muñoz V., Sánchez J., Equivariant vector bundles and logarithmic connections on toric varieties, J. Algebra 384 (2013), 227-241, arXiv:1207.6967.
  5. Biswas I., Senthamarai Kannan S., Nagaraj D.S., Equivariant principal bundles for $G$-actions and $G$-connections, Complex Manifolds 2 (2015), 178-185.
  6. Bruzzo U., Lie algebroid cohomology as a derived functor, J. Algebra 483 (2017), 245-261, arXiv:1606.02487.
  7. Bruzzo U., Mencattini I., Rubtsov V.N., Tortella P., Nonabelian holomorphic Lie algebroid extensions, Internat. J. Math. 26 (2015), 1550040, 26 pages, arXiv:1305.2377.
  8. Calaque D., Formality for Lie algebroids, Comm. Math. Phys. 257 (2005), 563-578, arXiv:math.QA/0404265.
  9. Calaque D., Rossi C.A., van den Bergh M., Hochschild (co)homology for Lie algebroids, Int. Math. Res. Not. 2010 (2010), 4098-4136, arXiv:0908.2630.
  10. Calaque D., van den Bergh M., Hochschild cohomology and Atiyah classes, Adv. Math. 224 (2010), 1839-1889, arXiv:0708.2725.
  11. Castro-Jiménez F.J., Narv'aez-Macarro L., Mond D., Cohomology of the complement of a free divisor, Trans. Amer. Math. Soc. 348 (1996), 3037-3049.
  12. Danilov V.I., Cohomology of algebraic varieties, in Algebraic Geometry, II, Encyclopaedia Math. Sci., Vol. 35, Springer, Berlin, 1996, 1-125.
  13. Dasgupta J., Khan B., Poddar M., Logarithmic connections on principal bundles over normal varieties, Bull. Sci. Math. 206 (2026), 103715, 35 pages, arXiv:2211.03047.
  14. Evens S., Lu J.-H., Weinstein A., Transverse measures, the modular class and a cohomology pairing for Lie algebroids, Quart. J. Math. Oxford Ser. (2) 50 (1999), 417-436, arXiv:dg-ga/9610008.
  15. Fernandes R.L., Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), 119-179, arXiv:math.DG/0007132.
  16. Forstnerič F., Stein manifolds and holomorphic mappings: The homotopy principle in complex analysis, Ergeb. Math. Grenzgeb. (3), Vol. 56, Springer, Heidelberg, 2011.
  17. Gerstenhaber M., The cohomology structure of an associative ring, Ann. of Math. 78 (1963), 267-288.
  18. Gerstenhaber M., Schack S.D., Algebras, bialgebras, quantum groups, and algebraic deformations, in Deformation Theory and Quantum Groups with Applications to Mathematical Physics (Amherst, MA, 1990), Contemp. Math., Vol. 134, American Mathematical Society, Providence, RI, 1992, 51-92.
  19. Gualtieri M., Generalized complex geometry, Ann. of Math. 174 (2011), 75-123.
  20. Hitchin N., Generalized holomorphic bundles and the $B$-field action, J. Geom. Phys. 61 (2011), 352-362, arXiv:1010.0207.
  21. Huebschmann J., Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990), 57-113, arXiv:1303.3903.
  22. Huebschmann J., Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras, Ann. Inst. Fourier (Grenoble) 48 (1998), 425-440, arXiv:dg-ga/9704005.
  23. Huebschmann J., Twilled Lie-Rinehart algebras and differential Batalin-Vilkovisky algebras, arXiv:math.DG/9811069.
  24. Huebschmann J., Duality for Lie-Rinehart algebras and the modular class, J. Reine Angew. Math. 510 (1999), 103-159, arXiv:dg-ga/9702008.
  25. Kapranov M., Free Lie algebroids and the space of paths, Selecta Math. (N.S.) 13 (2007), 277-319, arXiv:math.AG/0702584.
  26. Khan B., Poddar M., $G$-connections on principal bundles over complete $G$-varieties, J. Pure Appl. Algebra 229 (2025), 107816, 24 pages, arXiv:2402.00131.
  27. Klaasse R.L., Geometric structures and Lie algebroids, Ph.D. Thesis, Utrecht University, 2017, arXiv:1712.09560.
  28. Knighten C.M., Differentials on quotients of algebraic varieties, Trans. Amer. Math. Soc. 177 (1973), 65-89.
  29. Kosmann-Schwarzbach Y., Exact Gerstenhaber algebras and Lie bialgebroids, Acta Appl. Math. 41 (1995), 153-165.
  30. Kosmann-Schwarzbach Y., Magri F., Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), 35-81, available at http://www.numdam.org/item/AIHPA_1990__53_1_35_0.
  31. Laurent-Gengoux C., Louis R., Ryvkin L., What is a singular foliation?, in Advances in Poisson Geometry, Adv. Courses Math. CRM Barcelona, Birkhäuser, Cham, 2025, 123-261, arXiv:2407.14932.
  32. Laurent-Gengoux C., Ryvkin L., The neighborhood of a singular leaf, J. Éc. Polytech. Math. 8 (2021), 1037-1064, arXiv:2004.07019.
  33. Louis R., On Nash resolution of (singular) Lie algebroids, Math. Z. 311 (2025), 24, 27 pages, arXiv:2404.08840.
  34. Maakestad H.Ø., Algebraic connections on projective modules with prescribed curvature, J. Algebra 436 (2015), 161-227, arXiv:1401.7760.
  35. Mackenzie K.C.H., General theory of Lie groupoids and Lie algebroids, London Math. Soc. Lecture Note Ser., Vol. 213, Cambridge University Press, Cambridge, 2005.
  36. Mackenzie K.C.H., Xu P., Lie bialgebroids and Poisson groupoids, Duke Math. J. 73 (1994), 415-452, arXiv:dg-ga/9712012.
  37. Mandal A., Sarkar A., On Lie algebroid over algebraic spaces, Comm. Algebra 51 (2023), 1594-1613, arXiv:2107.11714.
  38. Moerdijk I., Mrčun J., On the universal enveloping algebra of a Lie algebroid, Proc. Amer. Math. Soc. 138 (2010), 3135-3145, arXiv:0801.3929.
  39. Narv'aez Macarro L., Differential structures in commutative algebra, Mini-course at the XXIII Brazilian Algebra Meeting (Maringá, Brazil, July 27 -- August 1, 2014), https://personal.us.es/narvaez/DSCA_course_2014.pdf.
  40. Nestruev J., Smooth manifolds and observables, Grad. Texts in Math., Vol. 220, Springer, Cham, 2020.
  41. Pym B., Poisson structures and Lie algebroids in complex geometry, Ph.D. Thesis, University of Toronto, 2013, https://www.math.mcgill.ca/bpym/files/bpym-thesis.pdf.
  42. Ramanan S., Global calculus, Grad. Stud. Math., Vol. 65, American Mathematical Society, Providence, RI, 2005.
  43. Rinehart G.S., Differential forms on general commutative algebras, Trans. Amer. Math. Soc. 108 (1963), 195-222.
  44. Sarkar A., Cohomology of Lie algebroids over algebraic spaces, arXiv:2111.01735.
  45. Schedler T., Deformations of algebras in noncommutative geometry, in Noncommutative Algebraic Geometry, Math. Sci. Res. Inst. Publ., Vol. 64, Cambridge University Press, New York, 2016, 71-165, arXiv:1212.0914.
  46. Tortella P., $\Lambda$-modules and holomorphic Lie algebroid connections, Cent. Eur. J. Math. 10 (2012), 1422-1441, arXiv:1108.3306.
  47. Tortella P., Representations of Atiyah algebroids and logarithmic connections, Int. Math. Res. Not. 2017 (2017), 29-46, arXiv:1505.04763.
  48. Villatoro J., On sheaves of Lie-Rinehart algebras, arXiv:2010.15463.
  49. Weibel C.A., Cyclic homology for schemes, Proc. Amer. Math. Soc. 124 (1996), 1655-1662.
  50. Weibel C.A., The $K$-book: An introduction to algebraic $K$-theory, Grad. Stud. Math., Vol. 145, American Mathematical Society, Providence, RI, 2013.
  51. Xu P., Gerstenhaber algebras and BV-algebras in Poisson geometry, Comm. Math. Phys. 200 (1999), 545-560, arXiv:dg-ga/9703001.
  52. Zambon M., Singular subalgebroids (with an appendix by Iakovos Androulidakis), Ann. Inst. Fourier (Grenoble) 72 (2022), 2109-2190, arXiv:1805.02480.

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