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December  2020, 12(4): 585-606. doi: 10.3934/jgm.2020025

Linearization of the higher analogue of Courant algebroids

Honglei Lang 1, and  Yunhe Sheng 2,,

1. 

Department of Applied Mathematics, China Agricultural University, Beijing, 100083, China
2. 

Department of Mathematics, Jilin University, Changchun, 130012, China

* Corresponding author: Yunhe Sheng

Received  February 2020 Revised  July 2020 Published  December 2020 Early access  September 2020

Fund Project: The first author is supported by NSFC grant 11901568; The second author is supported by NSFC grant 11922110

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.

Keywords: omni n-Lie algebroid, n-omni-Lie algebroid, higher analogue of Courant algebroids, linearization, n-Lie algebroid, Nambu-Jacobi structure.
Mathematics Subject Classification: Primary: 53D17, 53D18.
Citation: Honglei Lang, Yunhe Sheng. Linearization of the higher analogue of Courant algebroids. Journal of Geometric Mechanics, 2020, 12 (4) : 585-606. doi: 10.3934/jgm.2020025
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.

[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.

[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.

[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.

[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.

Table 1.   
omni $n$-Lie algebroids $n$-omni-Lie algebroids
$\mathfrak{D} E\oplus \wedge^n \mathfrak{J} E$ $\mathfrak{D} E\oplus \mathfrak{J}_n E$
Weinstein-linearization pseudo-linearization
$(n+1)$-Lie algebroid structures on $E$ higher Dirac-Jacobi structures
Nambu-Jacobi structures on $M$ exact multi-symplectic structures
Leibniz algebroid structures on $\wedge^n \mathfrak{J} E$ -
omni $n$-Lie algebra ${\rm{gl}}(V)\oplus \wedge^n V$ -
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omni $n$-Lie algebroids $n$-omni-Lie algebroids
$\mathfrak{D} E\oplus \wedge^n \mathfrak{J} E$ $\mathfrak{D} E\oplus \mathfrak{J}_n E$
Weinstein-linearization pseudo-linearization
$(n+1)$-Lie algebroid structures on $E$ higher Dirac-Jacobi structures
Nambu-Jacobi structures on $M$ exact multi-symplectic structures
Leibniz algebroid structures on $\wedge^n \mathfrak{J} E$ -
omni $n$-Lie algebra ${\rm{gl}}(V)\oplus \wedge^n V$ -
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