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

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

* 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.

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. BiL. 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. BursztynN. 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. ChenZ. Liu and Y. Sheng, $E$-Courant algebroids, Int. Math. Res. Not. IMRN, 2010 (2010), 4334-4376.  doi: 10.1093/imrn/rnq053.

[9]

Z. ChenZ. 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. GrabowskiD. 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. IbánezM de LeónJ. 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. IbánezB. LopezJ. 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-PonteC. 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. LangY. 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. LiuA. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547-574.  doi: 10.4310/jdg/1214459842.

[33]

G. MarmoG. 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. ShengZ. 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. BiL. 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. BursztynN. 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. ChenZ. Liu and Y. Sheng, $E$-Courant algebroids, Int. Math. Res. Not. IMRN, 2010 (2010), 4334-4376.  doi: 10.1093/imrn/rnq053.

[9]

Z. ChenZ. 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. GrabowskiD. 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. IbánezM de LeónJ. 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. IbánezB. LopezJ. 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-PonteC. 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. LangY. 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. LiuA. Weinstein and P. Xu, Manin triples for Lie bialgebroids, J. Differential Geom., 45 (1997), 547-574.  doi: 10.4310/jdg/1214459842.

[33]

G. MarmoG. 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. ShengZ. 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 ${\rm{gl}}(V)\oplus \wedge^n V$ -
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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