Extended curvatures and Lie pseudoalgebras
ITM Web of Conferences
https://doi.org/10.1051/ITMCONF/20224902007Abstract
The aim of the paper is to prove that if some extended curvatures on a preinfinitesimal module L, considered in the paper, vanish, then the derived preinfinitesimal module L(1) is a Lie pseudoalgebra. Two non-trivial examples are given. The first example is when L0 is an infinitesimal module and the second one is when L1 is a preinfinitesimal module.
References (26)
- Anastasiei M., Banach Lie algebroids and Dirac structures, Balk. J. Geom. Appl. 2013, 18.1, 1-11.
- Anastasiei M., Sandovici A., Banach Dirac bundles. Int. J. Geom. Meth. Mod. Phys. 2013, 10.07: 1350033.
- Balcerzak B., Symmetric brackets induced by connections with totally skew-symmetric torsion on skew-symmetric algebroids, arXiv preprint arXiv:2001.07390 (2020).
- Cabau P., Pelletier F., Almost Lie structures on an anchored Banach bundle, J. Geom. Phys. 2012, 62.11, 2147-2169.
- Chen Z., Liu Z-J., On (co-) morphisms of Lie pseudoalgebras and groupoids, J. Algebra 316.1 (2007): 1-31.
- Da Silva A.C., Weinstein A., Geometric models for noncommutative algebras, Vol. 10. AMS, 1999.
- Fernandes R.L., Lie algebroids, holonomy and characteristic classes, Adv. in Mathemat- ics 2002, 170.1, 119-179.
- Grabowski J., Brackets, Int. J. Geom. Meth. Mod. Phys. 2013, 10.08, 1360001.
- Grabowski J., and Jóźwikowski M., Pontryagin maximum principle on almost Lie alge- broids, SIAM J. Control Optim. 2011, 49.3 (2011): 1306-1357.
- Grabowski J., Khudaverdyan D., Poncin N.. The supergeometry of Loday algebroids, arXiv preprint arXiv:1103.5852 (2011).
- Grabowska K., Grabowski J., Dirac algebroids in Lagrangian and Hamiltonian mechan- ics, J. Geom. Phys. 2011, 61.11, 2233-2253.
- Huebschmann J., Lie-Rinehart algebras, Gerstenhaber algebras and Batalin-Vilkovisky algebras, Annales de l'institut Fourier 1998, 48.2.
- Mackenzie K., General theory of Lie groupoids and Lie algebroids, Cambridge Univ.Press., London, 2005.
- Mackenzie K., Lie algebroids and Lie pseudoalgebras, Bulletin of the London Mathe- matical Society 1995 27.2, 97-147.
- Marle C-M., Differential calculus on a Lie algebroids and Poisson manifolds, arXiv preprint. arXiv:0804.2451v2 (2008).
- Popescu M., Popescu P., Geometric objects defined by almost Lie structures, Banach Center Publications 2000, 54, 217-233.
- Popescu M., Popescu P., Almost Lie algebroids and characteristic classes, SIGMA 2019, 15, 021.
- Popescu M., Popescu P., Extended Curvatures and Lie Algebroids, Symmetry 2022, 14.7, 1375.
- Popescu P., On the geometry of relative tangent spaces, Rev. roum. math. pures appl. 1992, 37, 8, 727-733.
- Popescu P., Almost Lie structures, derivations and R-curvature on relative tangent spaces, Rev. roum. math. pures appl. 1992, 37, 9, 779-789.
- Popescu P., Categories of modules with differentials, J. Algebra 1996, 185, 50-73.
- Popescu P" The Lie pseudoalgebra of an anchored module, Algebra Colloquium 2006, 13.02.
- Popescu P., Popescu M., Anchored vector bundles and Lie algebroids, Banach Center Publications 2000, 54, 51-69.
- Popescu P., Poisson structures on almost complex Lie algebroids, Int. J. Geom. Meth. Mod. Phys. 2014, 11.08, 1450069.
- Pradines J., Théorie de Lie pour les groupoïdes di fférentiables. Calcul différentiel dans la cat égorie des groupoïdes infinitésimaux, C. R. Acad. Sci. Paris 1967, 264, 245-248.
- Vulcu V.-A., Dirac Structures on Banach Lie algebroids. Analele Stiintifice ale Univ. Ovidius Constanta, Ser. Mat. 2014, 22.3: 219-228.