Lie algebroids generated by cohomology operators

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September 2015, 7(3): 295-315. doi: 10.3934/jgm.2015.7.295

Lie algebroids generated by cohomology operators

Dennise García-Beltrán ^1, , José A. Vallejo ^2, and Yurii Vorobiev ^3,

1.	Departamento de Matemáticas, Universidad de Sonora, Blvd. Encinas y Rosales, Edicio 3K-1, Hermosillo, Son 83000, Mexico
2.	Facultad de Ciencias, Universidad Autónoma de San Luis Potosí, Lat. Av. Salvador Nava s/n Col. Lomas, San Luis Potosí, SLP 78290, Mexico
3.	Departamento de Matemáticas, Universidad de Sonora, Blvd. Encinas y Rosales, Edificio 3K-1, Hermosillo, Son 83000, Mexico

Received October 2014 Revised May 2015 Published July 2015

Abstract
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By studying the Frölicher-Nijenhuis decomposition of cohomology operators (that is, derivations $D$ of the exterior algebra $\Omega (M)$ with $\mathbb{Z}-$degree $1$ and $D^2=0$), we describe new examples of Lie algebroid structures on the tangent bundle $TM$ (and its complexification $T^{\mathbb{C}}M$) constructed from pre-existing geometric ones such as foliations, complex, product or tangent structures. We also describe a class of Lie algebroids on tangent bundles associated to idempotent endomorphisms with nontrivial Nijenhuis torsion.

Keywords: Lie algebroids, tangent structures, sprays., cohomology operators, complex structures, product structures.

Mathematics Subject Classification: Primary: 58J10, 53D17; Secondary: 20G1.

Citation: Dennise García-Beltrán, José A. Vallejo, Yurii Vorobiev. Lie algebroids generated by cohomology operators. Journal of Geometric Mechanics, 2015, 7 (3) : 295-315. doi: 10.3934/jgm.2015.7.295

References:

[1]	W. Ambrose, R. S. Palais and I. M. Singer, Sprays, An. Acad. Bras. Cie., 32 (1960), 163-178. Available from http://vmm.math.uci.edu/PalaisPapers/Sprays(withAmbrose&Singer).pdf.
[2]	A. D. Blaom, Geometric structures as deformed infinitesimal symmetries, Trans. Amer. Math. Soc., 358 (2006), 3651-3671. doi: 10.1090/S0002-9947-06-04057-8.
[3]	F. Cantrijn, J. Cariñena, J. Crampin and L. Ibort, Reduction of degenerate Lagrangian systems, J. Geom. Phys., 3 (1986), 353-400. doi: 10.1016/0393-0440(86)90014-8.
[4]	J. Clemente-Gallardo, Applications of Lie algebroids in mechanics and control theory, in Nonlinear control in the Year 2000, Lect. Not. in Control and Inf. Sci., 258 (2001), 299-313. doi: 10.1007/BFb0110222.
[5]	M. Crainic and R. L. Fernandes, Lectures on integrability of Lie brackets, Geometry and Topology Monographs, 17 (2011), 1-107. doi: 10.2140/gtm.2011.17.1.
[6]	M. Crainic and I. Moerdijk, Deformations of Lie brackets: Cohomological aspects, J. Eur. Math. Soc., 10 (2008), 1037-1059. doi: 10.4171/JEMS/139.
[7]	M. Crampin, On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics, J. Phys. A: Math. Gen., 14 (1981), 2567-2575. doi: 10.1088/0305-4470/14/10/012.
[8]	L. De Andrés, M. De León and P. R. Rodrigues, Connections on tangent bundles of higher order associated to regular Lagrangians, Geometriae Dedicata, 39 (1991), 17-28. doi: 10.1007/BF00147300.
[9]	M. De León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, 158, Elsevier, Amsterdam, 1998.
[10]	R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. in Math., 170 (2002), 119-179. doi: 10.1006/aima.2001.2070.
[11]	A. Frölicher and A. Nijenhuis, Theory of vector valued differential forms. Part I., Indagationes Math., 18 (1956), 338-359. doi: 10.1016/S1385-7258(56)50046-7.
[12]	J. Grabowski, Courant-Nijenhuis tensors and generalized geometries, in Groups, geometry and physics, Monografías de la Real Academia de Ciencias de Zaragoza, 29 (2006), 101-112. Available from: http://arxiv.org/abs/math/0601761.
[13]	J. Grabowski, Brackets, Int. J. of Geom. Methods in Mod. Phys., 10 (2013) 1360001 (45 pages). doi: 10.1142/S0219887813600013.
[14]	J. Grifone, Structure presque-tangente et connexions, Ann. Inst. Fourier, 22 (1972), 287-334. Available from: https://eudml.org/doc/74069. doi: 10.5802/aif.407.
[15]	D. Husemöller, M. Joachim, B. Jurčo and M. Schottenloher, Basic Bundle Theory and $K-$Cohomology Invariants, Lecture Notes in Physics, 726, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-74956-1.
[16]	I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, 2nd edition, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.
[17]	Y. Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures, Annales de l'I.H.P., section A, 53 (1990), 35-81. Available from: http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1990__53_1/AIHPA_1990__53_1_35_0/AIHPA_1990__53_1_35_0.pdf.
[18]	Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Annales de l'Institut Fourier, 46 (1996), 1243-1274. doi: 10.5802/aif.1547.
[19]	K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc. Lec. Notes, 213, Cambridge UP, Cambridge, 2005. doi: 10.2277/0521499283.
[20]	E. Martínez, Lie algebroids in classical mechanics and optimal control, SIGMA, 3 (2007), Paper 050, 17 pp. doi: 10.3842/SIGMA.2007.050.
[21]	P. W. Michor, Remarks on the Frölicher-Nijenhuis bracket, Differential geometry and its applications (Brno, 1986), 197-220, Math. Appl. (East European Ser.), 27, Reidel, Dordrecht, 1987. Available from: http://www.mat.univie.ac.at/~michor/froe-nij.pdf.
[22]	J. Monterde and A. Montesinos, Integral curves of derivations, Ann. of Global Anal. and Geom., 6 (1988), 177-189. doi: 10.1007/BF00133038.
[23]	A. Nijenhuis and R. Richardson, Deformation of Lie algebra structures, J. Math. Mech., 17 (1967), 89-105. doi: 10.1512/iumj.1968.17.17005.
[24]	A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields I, Indagationes Math., 17 (1955), 390-403. doi: 10.1016/S1385-7258(55)50054-0.
[25]	I. Vaisman, Cohomology and Differential Forms, Marcel Dekker Inc., New York, 1973.
[26]	A. Weinstein, Lagrangian Mechanics and Groupoids, in Mechanics Day, Fields Inst. Comm., 7, AMS Publishing (1996), 207-231. Available from: http://math.berkeley.edu/~alanw/Lagrangian.tex.
[27]	A. Weinstein, The Integration Problem for Complex Lie Algebroids, in From Geometry to Quantum Mechanics, Progress in Mathematics, 252, pp 93-109, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4530-4\_7.

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References:

[1]	W. Ambrose, R. S. Palais and I. M. Singer, Sprays, An. Acad. Bras. Cie., 32 (1960), 163-178. Available from http://vmm.math.uci.edu/PalaisPapers/Sprays(withAmbrose&Singer).pdf.
[2]	A. D. Blaom, Geometric structures as deformed infinitesimal symmetries, Trans. Amer. Math. Soc., 358 (2006), 3651-3671. doi: 10.1090/S0002-9947-06-04057-8.
[3]	F. Cantrijn, J. Cariñena, J. Crampin and L. Ibort, Reduction of degenerate Lagrangian systems, J. Geom. Phys., 3 (1986), 353-400. doi: 10.1016/0393-0440(86)90014-8.
[4]	J. Clemente-Gallardo, Applications of Lie algebroids in mechanics and control theory, in Nonlinear control in the Year 2000, Lect. Not. in Control and Inf. Sci., 258 (2001), 299-313. doi: 10.1007/BFb0110222.
[5]	M. Crainic and R. L. Fernandes, Lectures on integrability of Lie brackets, Geometry and Topology Monographs, 17 (2011), 1-107. doi: 10.2140/gtm.2011.17.1.
[6]	M. Crainic and I. Moerdijk, Deformations of Lie brackets: Cohomological aspects, J. Eur. Math. Soc., 10 (2008), 1037-1059. doi: 10.4171/JEMS/139.
[7]	M. Crampin, On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics, J. Phys. A: Math. Gen., 14 (1981), 2567-2575. doi: 10.1088/0305-4470/14/10/012.
[8]	L. De Andrés, M. De León and P. R. Rodrigues, Connections on tangent bundles of higher order associated to regular Lagrangians, Geometriae Dedicata, 39 (1991), 17-28. doi: 10.1007/BF00147300.
[9]	M. De León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, 158, Elsevier, Amsterdam, 1998.
[10]	R. L. Fernandes, Lie algebroids, holonomy and characteristic classes, Adv. in Math., 170 (2002), 119-179. doi: 10.1006/aima.2001.2070.
[11]	A. Frölicher and A. Nijenhuis, Theory of vector valued differential forms. Part I., Indagationes Math., 18 (1956), 338-359. doi: 10.1016/S1385-7258(56)50046-7.
[12]	J. Grabowski, Courant-Nijenhuis tensors and generalized geometries, in Groups, geometry and physics, Monografías de la Real Academia de Ciencias de Zaragoza, 29 (2006), 101-112. Available from: http://arxiv.org/abs/math/0601761.
[13]	J. Grabowski, Brackets, Int. J. of Geom. Methods in Mod. Phys., 10 (2013) 1360001 (45 pages). doi: 10.1142/S0219887813600013.
[14]	J. Grifone, Structure presque-tangente et connexions, Ann. Inst. Fourier, 22 (1972), 287-334. Available from: https://eudml.org/doc/74069. doi: 10.5802/aif.407.
[15]	D. Husemöller, M. Joachim, B. Jurčo and M. Schottenloher, Basic Bundle Theory and $K-$Cohomology Invariants, Lecture Notes in Physics, 726, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-74956-1.
[16]	I. Kolář, P. W. Michor and J. Slovák, Natural Operations in Differential Geometry, 2nd edition, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-662-02950-3.
[17]	Y. Kosmann-Schwarzbach and F. Magri, Poisson-Nijenhuis structures, Annales de l'I.H.P., section A, 53 (1990), 35-81. Available from: http://archive.numdam.org/ARCHIVE/AIHPA/AIHPA_1990__53_1/AIHPA_1990__53_1_35_0/AIHPA_1990__53_1_35_0.pdf.
[18]	Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras, Annales de l'Institut Fourier, 46 (1996), 1243-1274. doi: 10.5802/aif.1547.
[19]	K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Math. Soc. Lec. Notes, 213, Cambridge UP, Cambridge, 2005. doi: 10.2277/0521499283.
[20]	E. Martínez, Lie algebroids in classical mechanics and optimal control, SIGMA, 3 (2007), Paper 050, 17 pp. doi: 10.3842/SIGMA.2007.050.
[21]	P. W. Michor, Remarks on the Frölicher-Nijenhuis bracket, Differential geometry and its applications (Brno, 1986), 197-220, Math. Appl. (East European Ser.), 27, Reidel, Dordrecht, 1987. Available from: http://www.mat.univie.ac.at/~michor/froe-nij.pdf.
[22]	J. Monterde and A. Montesinos, Integral curves of derivations, Ann. of Global Anal. and Geom., 6 (1988), 177-189. doi: 10.1007/BF00133038.
[23]	A. Nijenhuis and R. Richardson, Deformation of Lie algebra structures, J. Math. Mech., 17 (1967), 89-105. doi: 10.1512/iumj.1968.17.17005.
[24]	A. Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields I, Indagationes Math., 17 (1955), 390-403. doi: 10.1016/S1385-7258(55)50054-0.
[25]	I. Vaisman, Cohomology and Differential Forms, Marcel Dekker Inc., New York, 1973.
[26]	A. Weinstein, Lagrangian Mechanics and Groupoids, in Mechanics Day, Fields Inst. Comm., 7, AMS Publishing (1996), 207-231. Available from: http://math.berkeley.edu/~alanw/Lagrangian.tex.
[27]	A. Weinstein, The Integration Problem for Complex Lie Algebroids, in From Geometry to Quantum Mechanics, Progress in Mathematics, 252, pp 93-109, Birkhäuser, Boston, 2007. doi: 10.1007/978-0-8176-4530-4\_7.

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