-
Previous Article
Models for higher algebroids
- JGM Home
- This Issue
-
Next Article
Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach
Lie algebroids generated by cohomology operators
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 |
References:
[1] |
W. Ambrose, R. S. Palais and I. M. Singer, Sprays,, An. Acad. Bras. Cie., 32 (1960), 163.
|
[2] |
A. D. Blaom, Geometric structures as deformed infinitesimal symmetries,, Trans. Amer. Math. Soc., 358 (2006), 3651.
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.
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, 258 (2001), 299.
doi: 10.1007/BFb0110222. |
[5] |
M. Crainic and R. L. Fernandes, Lectures on integrability of Lie brackets,, Geometry and Topology Monographs, 17 (2011), 1.
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.
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.
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.
doi: 10.1007/BF00147300. |
[9] |
M. De León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics,, North-Holland Mathematics Studies, 158 (1998). Google Scholar |
[10] |
R. L. Fernandes, Lie algebroids, holonomy and characteristic classes,, Adv. in Math., 170 (2002), 119.
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.
doi: 10.1016/S1385-7258(56)50046-7. |
[12] |
J. Grabowski, Courant-Nijenhuis tensors and generalized geometries,, in Groups, 29 (2006), 101.
|
[13] |
J. Grabowski, Brackets,, Int. J. of Geom. Methods in Mod. Phys., 10 (2013).
doi: 10.1142/S0219887813600013. |
[14] |
J. Grifone, Structure presque-tangente et connexions,, Ann. Inst. Fourier, 22 (1972), 287.
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 (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, (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., 53 (1990), 35.
|
[18] |
Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras,, Annales de l'Institut Fourier, 46 (1996), 1243.
doi: 10.5802/aif.1547. |
[19] |
K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, London Math. Soc. Lec. Notes, 213 (2005).
doi: 10.2277/0521499283. |
[20] |
E. Martínez, Lie algebroids in classical mechanics and optimal control,, SIGMA, 3 (2007).
doi: 10.3842/SIGMA.2007.050. |
[21] |
P. W. Michor, Remarks on the Frölicher-Nijenhuis bracket,, Differential geometry and its applications (Brno, (1986), 197.
|
[22] |
J. Monterde and A. Montesinos, Integral curves of derivations,, Ann. of Global Anal. and Geom., 6 (1988), 177.
doi: 10.1007/BF00133038. |
[23] |
A. Nijenhuis and R. Richardson, Deformation of Lie algebra structures,, J. Math. Mech., 17 (1967), 89.
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.
doi: 10.1016/S1385-7258(55)50054-0. |
[25] |
I. Vaisman, Cohomology and Differential Forms,, Marcel Dekker Inc., (1973).
|
[26] |
A. Weinstein, Lagrangian Mechanics and Groupoids,, in Mechanics Day, 7 (1996), 207.
|
[27] |
A. Weinstein, The Integration Problem for Complex Lie Algebroids,, in From Geometry to Quantum Mechanics, 252 (2007), 93.
doi: 10.1007/978-0-8176-4530-4\_7. |
show all references
References:
[1] |
W. Ambrose, R. S. Palais and I. M. Singer, Sprays,, An. Acad. Bras. Cie., 32 (1960), 163.
|
[2] |
A. D. Blaom, Geometric structures as deformed infinitesimal symmetries,, Trans. Amer. Math. Soc., 358 (2006), 3651.
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.
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, 258 (2001), 299.
doi: 10.1007/BFb0110222. |
[5] |
M. Crainic and R. L. Fernandes, Lectures on integrability of Lie brackets,, Geometry and Topology Monographs, 17 (2011), 1.
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.
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.
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.
doi: 10.1007/BF00147300. |
[9] |
M. De León and P. R. Rodrigues, Methods of Differential Geometry in Analytical Mechanics,, North-Holland Mathematics Studies, 158 (1998). Google Scholar |
[10] |
R. L. Fernandes, Lie algebroids, holonomy and characteristic classes,, Adv. in Math., 170 (2002), 119.
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.
doi: 10.1016/S1385-7258(56)50046-7. |
[12] |
J. Grabowski, Courant-Nijenhuis tensors and generalized geometries,, in Groups, 29 (2006), 101.
|
[13] |
J. Grabowski, Brackets,, Int. J. of Geom. Methods in Mod. Phys., 10 (2013).
doi: 10.1142/S0219887813600013. |
[14] |
J. Grifone, Structure presque-tangente et connexions,, Ann. Inst. Fourier, 22 (1972), 287.
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 (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, (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., 53 (1990), 35.
|
[18] |
Y. Kosmann-Schwarzbach, From Poisson algebras to Gerstenhaber algebras,, Annales de l'Institut Fourier, 46 (1996), 1243.
doi: 10.5802/aif.1547. |
[19] |
K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids,, London Math. Soc. Lec. Notes, 213 (2005).
doi: 10.2277/0521499283. |
[20] |
E. Martínez, Lie algebroids in classical mechanics and optimal control,, SIGMA, 3 (2007).
doi: 10.3842/SIGMA.2007.050. |
[21] |
P. W. Michor, Remarks on the Frölicher-Nijenhuis bracket,, Differential geometry and its applications (Brno, (1986), 197.
|
[22] |
J. Monterde and A. Montesinos, Integral curves of derivations,, Ann. of Global Anal. and Geom., 6 (1988), 177.
doi: 10.1007/BF00133038. |
[23] |
A. Nijenhuis and R. Richardson, Deformation of Lie algebra structures,, J. Math. Mech., 17 (1967), 89.
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.
doi: 10.1016/S1385-7258(55)50054-0. |
[25] |
I. Vaisman, Cohomology and Differential Forms,, Marcel Dekker Inc., (1973).
|
[26] |
A. Weinstein, Lagrangian Mechanics and Groupoids,, in Mechanics Day, 7 (1996), 207.
|
[27] |
A. Weinstein, The Integration Problem for Complex Lie Algebroids,, in From Geometry to Quantum Mechanics, 252 (2007), 93.
doi: 10.1007/978-0-8176-4530-4\_7. |
[1] |
Paulo Antunes, Joana M. Nunes da Costa. Hypersymplectic structures on Courant algebroids. Journal of Geometric Mechanics, 2015, 7 (3) : 255-280. doi: 10.3934/jgm.2015.7.255 |
[2] |
Melvin Leok, Diana Sosa. Dirac structures and Hamilton-Jacobi theory for Lagrangian mechanics on Lie algebroids. Journal of Geometric Mechanics, 2012, 4 (4) : 421-442. doi: 10.3934/jgm.2012.4.421 |
[3] |
Javier Pérez Álvarez. Invariant structures on Lie groups. Journal of Geometric Mechanics, 2020, 12 (2) : 141-148. doi: 10.3934/jgm.2020007 |
[4] |
Valentina Casarino, Paolo Ciatti, Silvia Secco. Product structures and fractional integration along curves in the space. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 619-635. doi: 10.3934/dcdss.2013.6.619 |
[5] |
Mohammad Shafiee. The 2-plectic structures induced by the Lie bialgebras. Journal of Geometric Mechanics, 2017, 9 (1) : 83-90. doi: 10.3934/jgm.2017003 |
[6] |
Pengliang Xu, Xiaomin Tang. Graded post-Lie algebra structures and homogeneous Rota-Baxter operators on the Schrödinger-Virasoro algebra. Electronic Research Archive, , () : -. doi: 10.3934/era.2021013 |
[7] |
Dennis I. Barrett, Rory Biggs, Claudiu C. Remsing, Olga Rossi. Invariant nonholonomic Riemannian structures on three-dimensional Lie groups. Journal of Geometric Mechanics, 2016, 8 (2) : 139-167. doi: 10.3934/jgm.2016001 |
[8] |
William D. Kalies, Konstantin Mischaikow, Robert C.A.M. Vandervorst. Lattice structures for attractors I. Journal of Computational Dynamics, 2014, 1 (2) : 307-338. doi: 10.3934/jcd.2014.1.307 |
[9] |
Javier de la Cruz, Michael Kiermaier, Alfred Wassermann, Wolfgang Willems. Algebraic structures of MRD codes. Advances in Mathematics of Communications, 2016, 10 (3) : 499-510. doi: 10.3934/amc.2016021 |
[10] |
Francesco Maddalena, Danilo Percivale, Franco Tomarelli. Adhesive flexible material structures. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 553-574. doi: 10.3934/dcdsb.2012.17.553 |
[11] |
Keizo Hasegawa. Complex moduli and pseudo-Kahler structures on three-dimensional compact complex solvmanifolds. Electronic Research Announcements, 2007, 14: 30-34. doi: 10.3934/era.2007.14.30 |
[12] |
V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153 |
[13] |
Dmitry Tamarkin. Quantization of Poisson structures on R^2. Electronic Research Announcements, 1997, 3: 119-120. |
[14] |
Matthias Liero, Alexander Mielke, Mark A. Peletier, D. R. Michiel Renger. On microscopic origins of generalized gradient structures. Discrete & Continuous Dynamical Systems - S, 2017, 10 (1) : 1-35. doi: 10.3934/dcdss.2017001 |
[15] |
Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355 |
[16] |
S. E. Pastukhova. Asymptotic analysis in elasticity problems on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (3) : 577-604. doi: 10.3934/nhm.2009.4.577 |
[17] |
Jiaquan Zhan, Fanyong Meng. Cores and optimal fuzzy communication structures of fuzzy games. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1187-1198. doi: 10.3934/dcdss.2019082 |
[18] |
Giovanni Rastelli, Manuele Santoprete. Canonoid and Poissonoid transformations, symmetries and biHamiltonian structures. Journal of Geometric Mechanics, 2015, 7 (4) : 483-515. doi: 10.3934/jgm.2015.7.483 |
[19] |
Guillermo Dávila-Rascón, Yuri Vorobiev. Hamiltonian structures for projectable dynamics on symplectic fiber bundles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (3) : 1077-1088. doi: 10.3934/dcds.2013.33.1077 |
[20] |
Daniel Guan. Classification of compact homogeneous spaces with invariant symplectic structures. Electronic Research Announcements, 1997, 3: 52-54. |
2019 Impact Factor: 0.649
Tools
Metrics
Other articles
by authors
[Back to Top]