American Institute of Mathematical Sciences

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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  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.  Google Scholar

[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.  Google Scholar

[3]

Y. Bi, L. Vitagliano and T. Zhang, Higher omni-Lie algebroids, J. Lie Theory, 29 (2019), 881-899.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Z. Chen and Z. Liu, Omni-Lie algebroids, J. Geom. Phys., 60 (2010), 799-808.  doi: 10.1016/j.geomphys.2010.01.007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

J. Grabowski, Brackets, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1360001, 45 pp. doi: 10.1142/S0219887813600013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Y. Hagiwara, Nambu-Dirac manifolds, J. Phys. A, 35 (2002), 1263-1281.  doi: 10.1088/0305-4470/35/5/310.  Google Scholar

[21]

Y. Hagiwara, Nambu-Jacobi structures and Jacobi algebroids, J. Phys. A, 37 (2004), 6713-6725.  doi: 10.1088/0305-4470/37/26/008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

K. Mikami and T. Mizutani, Foliations associated with Nambu-Jacobi structures, Tokyo J. Math., 28 (2005), 33-54.  doi: 10.3836/tjm/1244208277.  Google Scholar

[35]

Y. Sheng, On deformation of Lie algebroids, Results Math., 62 (2012), 103-120.  doi: 10.1007/s00025-011-0133-x.  Google Scholar

[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.  Google Scholar

[37]

K. Uchino, Courant brackets on noncommutative algebras and omni-Lie algebras, Tokyo J. Math., 30 (2007), 239-255.  doi: 10.3836/tjm/1184963659.  Google Scholar

[38]

L. Vitagliano, Dirac-Jacobi bundles, J. Symplectic Geom., 16 (2018), 485-561.  doi: 10.4310/JSG.2018.v16.n2.a4.  Google Scholar

[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.  Google Scholar

[40]

A. Wade, Conformal Dirac structures, Lett. Math. Phys., 53 (2000), 331-348.  doi: 10.1023/A:1007634407701.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[3]

Y. Bi, L. Vitagliano and T. Zhang, Higher omni-Lie algebroids, J. Lie Theory, 29 (2019), 881-899.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

Z. Chen and Z. Liu, Omni-Lie algebroids, J. Geom. Phys., 60 (2010), 799-808.  doi: 10.1016/j.geomphys.2010.01.007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[15]

J. Grabowski, Brackets, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1360001, 45 pp. doi: 10.1142/S0219887813600013.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

Y. Hagiwara, Nambu-Dirac manifolds, J. Phys. A, 35 (2002), 1263-1281.  doi: 10.1088/0305-4470/35/5/310.  Google Scholar

[21]

Y. Hagiwara, Nambu-Jacobi structures and Jacobi algebroids, J. Phys. A, 37 (2004), 6713-6725.  doi: 10.1088/0305-4470/37/26/008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

K. Mikami and T. Mizutani, Foliations associated with Nambu-Jacobi structures, Tokyo J. Math., 28 (2005), 33-54.  doi: 10.3836/tjm/1244208277.  Google Scholar

[35]

Y. Sheng, On deformation of Lie algebroids, Results Math., 62 (2012), 103-120.  doi: 10.1007/s00025-011-0133-x.  Google Scholar

[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.  Google Scholar

[37]

K. Uchino, Courant brackets on noncommutative algebras and omni-Lie algebras, Tokyo J. Math., 30 (2007), 239-255.  doi: 10.3836/tjm/1184963659.  Google Scholar

[38]

L. Vitagliano, Dirac-Jacobi bundles, J. Symplectic Geom., 16 (2018), 485-561.  doi: 10.4310/JSG.2018.v16.n2.a4.  Google Scholar

[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.  Google Scholar

[40]

A. Wade, Conformal Dirac structures, Lett. Math. Phys., 53 (2000), 331-348.  doi: 10.1023/A:1007634407701.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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