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

SIGMA 21 (2025), 044, 25 pages      arXiv:2401.05274      https://doi.org/10.3842/SIGMA.2025.044

On Complex Lie Algebroids with Constant Real Rank

Dan Aguero
Scuola Internazionale Superiore di Studi Avanzati - SISSA, Via Bonomea, 265, 34136 Trieste, Italy

Received September 30, 2024, in final form June 02, 2025; Published online June 13, 2025

Abstract
We associate a real distribution to any complex Lie algebroid that we call distribution of real elements and a new invariant that we call real rank, given by the pointwise rank of this distribution. When the real rank is constant, we obtain a real Lie algebroid inside the original complex Lie algebroid. Under another regularity condition, we associate a complex Lie subalgebroid that we call the minimal complex subalgebroid. We also provide a local splitting for complex Lie algebroids with constant real rank. In the last part, we introduce the complex matched pair of skew-algebroids; these pairs produce complex Lie algebroid structures on the complexification of a vector bundle. We use this operation to characterize all the complex Lie algebroid structures on the complexification of real vector bundles.

Key words: complex Lie algebroids; Poisson geometry; normal forms.

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References

  1. Aguero D., Complex Dirac structures with constant real index, Ph.D. Thesis, Instituto de Matematica Pura e Aplicada, 2020.
  2. Aguero D., Rubio R., Complex Dirac structures: invariants and local structure, Comm. Math. Phys. 396 (2022), 623-646, arXiv:2105.05265.
  3. Bailey M., Local classification of generalized complex structures, J. Differential Geom. 95 (2013), 1-37, arXiv:1201.4887.
  4. Bischoff F., Bursztyn H., Lima H., Meinrenken E., Deformation spaces and normal forms around transversals, Compos. Math. 156 (2020), 697-732, arXiv:1807.11153.
  5. Bursztyn H., Lima H., Meinrenken E., Splitting theorems for Poisson and related structures, J. Reine Angew. Math. 754 (2019), 281-312, arXiv:1605.05386.
  6. Cannas da Silva A., Weinstein A., Geometric models for noncommutative algebras, Berkeley Math. Lect. Notes, Vol. 10, American Mathematical Society, Providence, RI, 1999.
  7. Crainic M., Fernandes R.L., Integrability of Lie brackets, Ann. of Math. 157 (2003), 575-620, arXiv:math.DG/0105033.
  8. Dufour J.-P., Normal forms for Lie algebroids, in Lie Algebroids and Related Topics in Differential Geometry (Warsaw, 2000), Banach Center Publ., Vol. 54, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 2001, 35-41.
  9. Fernandes R.L., Lie algebroids, holonomy and characteristic classes, Adv. Math. 170 (2002), 119-179, arXiv:math.DG/0007132.
  10. Grabowski J., Ravanpak Z., Nonassociative analogs of Lie groupoids, Differential Geom. Appl. 82 (2022), 101887, 32 pages, arXiv:2101.07258.
  11. Gualtieri M., Generalized complex geometry, Ph.D. Thesis, Oxford University, 2004, arXiv:math.DG/0401221.
  12. Gualtieri M., Generalized complex geometry, Ann. of Math. 174 (2011), 75-123, arXiv:math.DG/0703298.
  13. Hitchin N., Generalized Calabi-Yau manifolds, Q. J. Math. 54 (2003), 281-308, arXiv:math.DG/0209099.
  14. Kosmann-Schwarzbach Y., Magri F., Poisson-Nijenhuis structures, Ann. Inst. H. Poincaré Phys. Théor. 53 (1990), 35-81.
  15. Laurent-Gengoux C., Stiénon M., Xu P., Holomorphic Poisson manifolds and holomorphic Lie algebroids, Int. Math. Res. Not. 2008 (2008), 088, 46 pages, arXiv:0707.4253.
  16. Laurent-Gengoux C., Stiénon M., Xu P., Integration of holomorphic Lie algebroids, Math. Ann. 345 (2009), 895-923, arXiv:0803.2031.
  17. Milnor J.W., Stasheff J.D., Characteristic classes, Ann. of Math. Stud., Vol. 76, Princeton University Press, Princeton, NJ, 1974.
  18. Mokri T., Matched pairs of Lie algebroids, Glasgow Math. J. 39 (1997), 167-181.
  19. Nirenberg L., A complex Frobenius theorem, Courant Institute of Mathematical Sciences, New York University, 1958.
  20. Pradines J., Troisième théorème de Lie les groupoïdes différentiables, C. R. Acad. Sci. Paris Sér. A-B 267 (1968), A21-A23.
  21. Rubio R., Generalized geometry of type $B_n$, Ph.D. Thesis, University of Oxford, 2014.
  22. Tennyson D., Waldram D., Exceptional complex structures and the hypermultiplet moduli of 5d Minkowski compactifications of M-theory, J. High Energy Phys. 2021 (2021), no. 8, 088, 62 pages, arXiv:2104.09900.
  23. Trèves F., Hypo-analytic structures. Local theory, Princeton Math. Ser., Vol. 40, Princeton University Press, Princeton, NJ, 1992.
  24. Weinstein A., The integration problem for complex Lie algebroids, in From Geometry to Quantum Mechanics, Progr. Math., Vol. 252, Birkhäuser, Boston, MA, 2007, 93-109, arXiv:math.DG/0601752.

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