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Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems
Linearization of the higher analogue of Courant algebroids
1. | Department of Applied Mathematics, China Agricultural University, Beijing, 100083, China |
2. | Department of Mathematics, Jilin University, Changchun, 130012, China |
In this paper, we show that the spaces of sections of the $ n $-th differential operator bundle $ \mathfrak{D}^n E $ and the $ n $-th skew-symmetric jet bundle $ \mathfrak{J}_n E $ of a vector bundle $ E $ are isomorphic to the spaces of linear $ n $-vector fields and linear $ n $-forms on $ E^* $ respectively. Consequently, the $ n $-omni-Lie algebroid $ \mathfrak{D} E\oplus \mathfrak{J}_n E $ introduced by Bi-Vitagliano-Zhang can be explained as certain linearization, which we call pseudo-linearization of the higher analogue of Courant algebroids $ TE^*\oplus \wedge^nT^*E^* $. On the other hand, we show that the omni $ n $-Lie algebroid $ \mathfrak{D} E\oplus \wedge^n \mathfrak{J} E $ can also be explained as certain linearization, which we call Weinstein-linearization of the higher analogue of Courant algebroids $ TE^*\oplus \wedge^nT^*E^* $. We also show that $ n $-Lie algebroids, local $ n $-Lie algebras and Nambu-Jacobi structures can be characterized as integrable subbundles of omni $ n $-Lie algebroids.
References:
[1] |
Y. Bi and Y. Sheng,
On higher analogues of Courant algebroids, Sci. China Math., 54 (2011), 437-447.
doi: 10.1007/s11425-010-4142-0. |
[2] |
Y. Bi and Y. Sheng, Dirac structures for higher analogues of Courant algebroids, Int. J. Geom. Methods Mod. Phys., 12 (2015), 1550010, 13 pp.
doi: 10.1142/S0219887815500103. |
[3] |
Y. Bi, L. Vitagliano and T. Zhang,
Higher omni-Lie algebroids, J. Lie Theory, 29 (2019), 881-899.
|
[4] |
P. Bouwknegt and B. Jurčo, AKSZ construction of topological open $p$-brane action and Nambu brackets, Rev. Math. Phys., 25 (2013), 1330004, 31 pp.
doi: 10.1142/S0129055X13300045. |
[5] |
H. Bursztyn and A. Cabrera,
Multiplicative forms at the infinitesimal level, Math. Ann., 353 (2012), 663-705.
doi: 10.1007/s00208-011-0697-5. |
[6] |
H. Bursztyn, N. Martinez Alba and R. Rubio,
On higher Dirac structures, Int. Math. Res. Not. IMRN, 2019 (2019), 1503-1542.
doi: 10.1093/imrn/rnx163. |
[7] |
Z. Chen and Z. Liu,
Omni-Lie algebroids, J. Geom. Phys., 60 (2010), 799-808.
doi: 10.1016/j.geomphys.2010.01.007. |
[8] |
Z. Chen, Z. Liu and Y. Sheng,
$E$-Courant algebroids, Int. Math. Res. Not. IMRN, 2010 (2010), 4334-4376.
doi: 10.1093/imrn/rnq053. |
[9] |
Z. Chen, Z. Liu and Y. Sheng,
Dirac structures of omni-Lie algebroids, Internat. J. Math., 22 (2011), 1163-1185.
doi: 10.1142/S0129167X11007215. |
[10] |
M. Crainic and I. Moerdijk,
Deformations of Lie brackets: Cohomological aspects, J. Eur. Math. Soc. (JEMS), 10 (2008), 1037-1059.
doi: 10.4171/JEMS/139. |
[11] |
M. Cueca, The geometry of graded cotangent bundles, preprint, arXiv: 1905.13245. Google Scholar |
[12] |
Y. Daletskii and L. Takhtajan,
Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys., 39 (1997), 127-141.
doi: 10.1023/A:1007316732705. |
[13] |
J. A. de Azc$\rm\acute{a}$rraga and J. M. Izquierdo, $n$-ary algebras: a review with applications, J. Phys. A: Math. Theor., 43 (2010), 293001. Google Scholar |
[14] |
V. T. Filippov,
$n$-Lie algebras, Sibirsk. Mat. Zh., 26 (1985), 126-140.
|
[15] |
J. Grabowski, Brackets, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1360001, 45 pp.
doi: 10.1142/S0219887813600013. |
[16] |
J. Grabowski, D. Khudaverdyan and N. Poncin,
The supergeometry of Loday algebroids, J. Geom. Mech., 5 (2013), 185-213.
doi: 10.3934/jgm.2013.5.185. |
[17] |
J. Grabowski and G. Marmo,
On Filippov algebroids and multiplicative Nambu-Poisson structures, Differential Geom. Appl., 12 (2000), 35-50.
doi: 10.1016/S0926-2245(99)00042-X. |
[18] |
J. Grabowski and M. Rotkiewicz,
Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.
doi: 10.1016/j.geomphys.2009.06.009. |
[19] |
M. Grutzmann and T. Strobl, General Yang-Mills type gauge theories for $p$-form gauge fields: From physics-based ideas to a mathematical framework or from Bianchi identities to twisted Courant algebroids, Int. J. Geom. Methods Mod. Phys., 12 (2015), 1550009, 80 pp.
doi: 10.1142/S0219887815500097. |
[20] |
Y. Hagiwara,
Nambu-Dirac manifolds, J. Phys. A, 35 (2002), 1263-1281.
doi: 10.1088/0305-4470/35/5/310. |
[21] |
Y. Hagiwara,
Nambu-Jacobi structures and Jacobi algebroids, J. Phys. A, 37 (2004), 6713-6725.
doi: 10.1088/0305-4470/37/26/008. |
[22] |
C. M. Hull, Generalised geometry for M-theory, J. High Energy Phys., 2007 (2007), 079, 31 pp.
doi: 10.1088/1126-6708/2007/07/079. |
[23] |
R. Ibánez, M de León, J. C. Marrero and E. Padrón,
Leibniz algebroid associated with a Nambu-Poisson structure, J. Phys. A, 32 (1999), 8129-8144.
doi: 10.1088/0305-4470/32/46/310. |
[24] |
R. Ibánez, B. Lopez, J. C. Marrero and E. Padrón,
Matched pairs of Leibniz algebroids, Nambu-Jacobi structures and modular class, C. R. Acad. Sci., Paris Sér I Math., 333 (2001), 861-866.
doi: 10.1016/S0764-4442(01)02150-4. |
[25] |
D. Iglesias-Ponte, C. Laurent-Gengoux and P. Xu,
Universal lifting theorem and quasi-Poisson groupoids, J. Eur. Math. Soc. (JEMS), 14 (2012), 681-731.
doi: 10.4171/JEMS/315. |
[26] |
D. Iglesias-Ponte and A. Wade,
Contact manifolds and generalized complex structures, J. Geom. Phys., 53 (2005), 249-258.
doi: 10.1016/j.geomphys.2004.06.006. |
[27] |
M. K. Kinyon and A. Weinstein,
Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Amer. J. Math., 123 (2001), 525-550.
doi: 10.1353/ajm.2001.0017. |
[28] |
Y. Kosmann-Schwarzbach, Courant algebroids. A short history, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), 014, 8 pp.
doi: 10.3842/SIGMA.2013.014. |
[29] |
P. P. La Pastina and L. Vitagliano,
Deformations of linear Lie brackets, Pacific J. Math., 303 (2019), 265-298.
doi: 10.2140/pjm.2019.303.265. |
[30] |
H. Lang, Y. Sheng and A. Wade,
VB-Courant algebroids, $E$-Courant algebroids and generalized geometry, Canad. Math. Bull., 61 (2018), 588-607.
doi: 10.4153/CMB-2017-079-7. |
[31] |
J. Liu, Y. Sheng and C. Wang, Omni $n$-Lie algebras and linearization of higher analogues of Courant algebroids, Int. J. Geom. Methods Mod. Phys., 14 (2017), 1750113, 18 pp.
doi: 10.1142/S0219887817501134. |
[32] |
Z. Liu, A. Weinstein and P. Xu,
Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547-574.
doi: 10.4310/jdg/1214459842. |
[33] |
G. Marmo, G. Vilasi and A. M. Vinogradov,
The local structure of $n$-Poisson and $n$-Jacobi manifolds, J. Geom. Phys., 25 (1998), 141-182.
doi: 10.1016/S0393-0440(97)00057-0. |
[34] |
K. Mikami and T. Mizutani,
Foliations associated with Nambu-Jacobi structures, Tokyo J. Math., 28 (2005), 33-54.
doi: 10.3836/tjm/1244208277. |
[35] |
Y. Sheng,
On deformation of Lie algebroids, Results Math., 62 (2012), 103-120.
doi: 10.1007/s00025-011-0133-x. |
[36] |
Y. Sheng, Z. Liu and C. Zhu,
Omni-Lie 2-algebras and their Dirac structures, J. Geom. Phys., 61 (2011), 560-575.
doi: 10.1016/j.geomphys.2010.11.005. |
[37] |
K. Uchino,
Courant brackets on noncommutative algebras and omni-Lie algebras, Tokyo J. Math., 30 (2007), 239-255.
doi: 10.3836/tjm/1184963659. |
[38] |
L. Vitagliano,
Dirac-Jacobi bundles, J. Symplectic Geom., 16 (2018), 485-561.
doi: 10.4310/JSG.2018.v16.n2.a4. |
[39] |
L. Vitagliano and A. Wade,
Generalized contact bundles, C. R. Math. Acad. Sci. Paris, 354 (2016), 313-317.
doi: 10.1016/j.crma.2015.12.009. |
[40] |
A. Wade,
Conformal Dirac structures, Lett. Math. Phys., 53 (2000), 331-348.
doi: 10.1023/A:1007634407701. |
[41] |
A. Weinstein, Omni-Lie algebras, Microlocal analysis of the Schrodinger equation and related topics (Japanese) (Kyoto, 1999), S${\bar{u}}$rikaisekikenky${\bar{u}}$sho K${\bar{u}}$ky${\bar{u}}$roku, 1176 (2000), 95–102. |
[42] |
M. Zambon,
$L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids, J. Symplectic Geom., 10 (2012), 563-599.
doi: 10.4310/JSG.2012.v10.n4.a4. |
show all references
References:
[1] |
Y. Bi and Y. Sheng,
On higher analogues of Courant algebroids, Sci. China Math., 54 (2011), 437-447.
doi: 10.1007/s11425-010-4142-0. |
[2] |
Y. Bi and Y. Sheng, Dirac structures for higher analogues of Courant algebroids, Int. J. Geom. Methods Mod. Phys., 12 (2015), 1550010, 13 pp.
doi: 10.1142/S0219887815500103. |
[3] |
Y. Bi, L. Vitagliano and T. Zhang,
Higher omni-Lie algebroids, J. Lie Theory, 29 (2019), 881-899.
|
[4] |
P. Bouwknegt and B. Jurčo, AKSZ construction of topological open $p$-brane action and Nambu brackets, Rev. Math. Phys., 25 (2013), 1330004, 31 pp.
doi: 10.1142/S0129055X13300045. |
[5] |
H. Bursztyn and A. Cabrera,
Multiplicative forms at the infinitesimal level, Math. Ann., 353 (2012), 663-705.
doi: 10.1007/s00208-011-0697-5. |
[6] |
H. Bursztyn, N. Martinez Alba and R. Rubio,
On higher Dirac structures, Int. Math. Res. Not. IMRN, 2019 (2019), 1503-1542.
doi: 10.1093/imrn/rnx163. |
[7] |
Z. Chen and Z. Liu,
Omni-Lie algebroids, J. Geom. Phys., 60 (2010), 799-808.
doi: 10.1016/j.geomphys.2010.01.007. |
[8] |
Z. Chen, Z. Liu and Y. Sheng,
$E$-Courant algebroids, Int. Math. Res. Not. IMRN, 2010 (2010), 4334-4376.
doi: 10.1093/imrn/rnq053. |
[9] |
Z. Chen, Z. Liu and Y. Sheng,
Dirac structures of omni-Lie algebroids, Internat. J. Math., 22 (2011), 1163-1185.
doi: 10.1142/S0129167X11007215. |
[10] |
M. Crainic and I. Moerdijk,
Deformations of Lie brackets: Cohomological aspects, J. Eur. Math. Soc. (JEMS), 10 (2008), 1037-1059.
doi: 10.4171/JEMS/139. |
[11] |
M. Cueca, The geometry of graded cotangent bundles, preprint, arXiv: 1905.13245. Google Scholar |
[12] |
Y. Daletskii and L. Takhtajan,
Leibniz and Lie algebra structures for Nambu algebra, Lett. Math. Phys., 39 (1997), 127-141.
doi: 10.1023/A:1007316732705. |
[13] |
J. A. de Azc$\rm\acute{a}$rraga and J. M. Izquierdo, $n$-ary algebras: a review with applications, J. Phys. A: Math. Theor., 43 (2010), 293001. Google Scholar |
[14] |
V. T. Filippov,
$n$-Lie algebras, Sibirsk. Mat. Zh., 26 (1985), 126-140.
|
[15] |
J. Grabowski, Brackets, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1360001, 45 pp.
doi: 10.1142/S0219887813600013. |
[16] |
J. Grabowski, D. Khudaverdyan and N. Poncin,
The supergeometry of Loday algebroids, J. Geom. Mech., 5 (2013), 185-213.
doi: 10.3934/jgm.2013.5.185. |
[17] |
J. Grabowski and G. Marmo,
On Filippov algebroids and multiplicative Nambu-Poisson structures, Differential Geom. Appl., 12 (2000), 35-50.
doi: 10.1016/S0926-2245(99)00042-X. |
[18] |
J. Grabowski and M. Rotkiewicz,
Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.
doi: 10.1016/j.geomphys.2009.06.009. |
[19] |
M. Grutzmann and T. Strobl, General Yang-Mills type gauge theories for $p$-form gauge fields: From physics-based ideas to a mathematical framework or from Bianchi identities to twisted Courant algebroids, Int. J. Geom. Methods Mod. Phys., 12 (2015), 1550009, 80 pp.
doi: 10.1142/S0219887815500097. |
[20] |
Y. Hagiwara,
Nambu-Dirac manifolds, J. Phys. A, 35 (2002), 1263-1281.
doi: 10.1088/0305-4470/35/5/310. |
[21] |
Y. Hagiwara,
Nambu-Jacobi structures and Jacobi algebroids, J. Phys. A, 37 (2004), 6713-6725.
doi: 10.1088/0305-4470/37/26/008. |
[22] |
C. M. Hull, Generalised geometry for M-theory, J. High Energy Phys., 2007 (2007), 079, 31 pp.
doi: 10.1088/1126-6708/2007/07/079. |
[23] |
R. Ibánez, M de León, J. C. Marrero and E. Padrón,
Leibniz algebroid associated with a Nambu-Poisson structure, J. Phys. A, 32 (1999), 8129-8144.
doi: 10.1088/0305-4470/32/46/310. |
[24] |
R. Ibánez, B. Lopez, J. C. Marrero and E. Padrón,
Matched pairs of Leibniz algebroids, Nambu-Jacobi structures and modular class, C. R. Acad. Sci., Paris Sér I Math., 333 (2001), 861-866.
doi: 10.1016/S0764-4442(01)02150-4. |
[25] |
D. Iglesias-Ponte, C. Laurent-Gengoux and P. Xu,
Universal lifting theorem and quasi-Poisson groupoids, J. Eur. Math. Soc. (JEMS), 14 (2012), 681-731.
doi: 10.4171/JEMS/315. |
[26] |
D. Iglesias-Ponte and A. Wade,
Contact manifolds and generalized complex structures, J. Geom. Phys., 53 (2005), 249-258.
doi: 10.1016/j.geomphys.2004.06.006. |
[27] |
M. K. Kinyon and A. Weinstein,
Leibniz algebras, Courant algebroids, and multiplications on reductive homogeneous spaces, Amer. J. Math., 123 (2001), 525-550.
doi: 10.1353/ajm.2001.0017. |
[28] |
Y. Kosmann-Schwarzbach, Courant algebroids. A short history, SIGMA Symmetry Integrability Geom. Methods Appl., 9 (2013), 014, 8 pp.
doi: 10.3842/SIGMA.2013.014. |
[29] |
P. P. La Pastina and L. Vitagliano,
Deformations of linear Lie brackets, Pacific J. Math., 303 (2019), 265-298.
doi: 10.2140/pjm.2019.303.265. |
[30] |
H. Lang, Y. Sheng and A. Wade,
VB-Courant algebroids, $E$-Courant algebroids and generalized geometry, Canad. Math. Bull., 61 (2018), 588-607.
doi: 10.4153/CMB-2017-079-7. |
[31] |
J. Liu, Y. Sheng and C. Wang, Omni $n$-Lie algebras and linearization of higher analogues of Courant algebroids, Int. J. Geom. Methods Mod. Phys., 14 (2017), 1750113, 18 pp.
doi: 10.1142/S0219887817501134. |
[32] |
Z. Liu, A. Weinstein and P. Xu,
Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547-574.
doi: 10.4310/jdg/1214459842. |
[33] |
G. Marmo, G. Vilasi and A. M. Vinogradov,
The local structure of $n$-Poisson and $n$-Jacobi manifolds, J. Geom. Phys., 25 (1998), 141-182.
doi: 10.1016/S0393-0440(97)00057-0. |
[34] |
K. Mikami and T. Mizutani,
Foliations associated with Nambu-Jacobi structures, Tokyo J. Math., 28 (2005), 33-54.
doi: 10.3836/tjm/1244208277. |
[35] |
Y. Sheng,
On deformation of Lie algebroids, Results Math., 62 (2012), 103-120.
doi: 10.1007/s00025-011-0133-x. |
[36] |
Y. Sheng, Z. Liu and C. Zhu,
Omni-Lie 2-algebras and their Dirac structures, J. Geom. Phys., 61 (2011), 560-575.
doi: 10.1016/j.geomphys.2010.11.005. |
[37] |
K. Uchino,
Courant brackets on noncommutative algebras and omni-Lie algebras, Tokyo J. Math., 30 (2007), 239-255.
doi: 10.3836/tjm/1184963659. |
[38] |
L. Vitagliano,
Dirac-Jacobi bundles, J. Symplectic Geom., 16 (2018), 485-561.
doi: 10.4310/JSG.2018.v16.n2.a4. |
[39] |
L. Vitagliano and A. Wade,
Generalized contact bundles, C. R. Math. Acad. Sci. Paris, 354 (2016), 313-317.
doi: 10.1016/j.crma.2015.12.009. |
[40] |
A. Wade,
Conformal Dirac structures, Lett. Math. Phys., 53 (2000), 331-348.
doi: 10.1023/A:1007634407701. |
[41] |
A. Weinstein, Omni-Lie algebras, Microlocal analysis of the Schrodinger equation and related topics (Japanese) (Kyoto, 1999), S${\bar{u}}$rikaisekikenky${\bar{u}}$sho K${\bar{u}}$ky${\bar{u}}$roku, 1176 (2000), 95–102. |
[42] |
M. Zambon,
$L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids, J. Symplectic Geom., 10 (2012), 563-599.
doi: 10.4310/JSG.2012.v10.n4.a4. |
omni |
|
Weinstein-linearization | pseudo-linearization |
higher Dirac-Jacobi structures | |
Nambu-Jacobi structures on |
exact multi-symplectic structures |
Leibniz algebroid structures on |
- |
omni |
- |
omni |
|
Weinstein-linearization | pseudo-linearization |
higher Dirac-Jacobi structures | |
Nambu-Jacobi structures on |
exact multi-symplectic structures |
Leibniz algebroid structures on |
- |
omni |
- |
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