June  2016, 8(2): 199-220. doi: 10.3934/jgm.2016004

Infinitesimally natural principal bundles

1. 

Universiteit Utrecht, Budapestlaan 6, 3584 CD Utrecht, Netherlands

Received  June 2015 Revised  April 2016 Published  June 2016

We extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. We classify the principal fibre bundles with this property. A version of the main result in this paper (theorem 4.4) can be found in Lecomte's work [12]. Our approach was developed independently, uses the language of Lie algebroids, and can be generalized in several directions.
Citation: Bas Janssens. Infinitesimally natural principal bundles. Journal of Geometric Mechanics, 2016, 8 (2) : 199-220. doi: 10.3934/jgm.2016004
References:
[1]

A. Banyaga, The Structure of Classical Diffeomorphism Groups, volume 400 of Mathematics and its Applications,, Kluwer Academic Publishers Group, (1997).  doi: 10.1007/978-1-4757-6800-8.  Google Scholar

[2]

D. W. Barnes, Nilpotency of Lie algebras,, Math. Zeitschr., 79 (1962), 237.  doi: 10.1007/BF01193118.  Google Scholar

[3]

M. Berg, C. DeWitt-Morette, S. Gwo and E. Kramer, The pin groups in physics: C, P and T,, Rev. Math. Phys., 13 (2001), 953.  doi: 10.1142/S0129055X01000922.  Google Scholar

[4]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Ann. of Math., 157 (2003), 575.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[5]

D. B. A. Epstein and W. P. Thurston, Transformation groups and natural bundles,, Proc. London Math. Soc., 38 (1979), 219.  doi: 10.1112/plms/s3-38.2.219.  Google Scholar

[6]

D. B. Fuks, Cohomology of Infinite Dimensional Lie Algebras,, Contemporary Soviet Mathematics, (1986).   Google Scholar

[7]

H. Glöckner, Differentiable mappings between spaces of sections, 2002., arXiv:1308.1172., ().   Google Scholar

[8]

J. Grabowski, A. Kotov and N. Poncin, Geometric structures encoded in the Lie structure of an Atiyah algebroid,, Transform. Groups, 16 (2011), 137.  doi: 10.1007/s00031-011-9126-9.  Google Scholar

[9]

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory,, Graduate Texts in Mathematics, (1972).   Google Scholar

[10]

A. W. Knapp, Lie Groups Beyond an Introduction,, Birkhäuser, (1996).  doi: 10.1007/978-1-4757-2453-0.  Google Scholar

[11]

H. B. Lawson and M.-L. Michelsohn, Spin geometry,, Princeton University Press, (1994).   Google Scholar

[12]

P. B. A. Lecomte, Sur la suite exacte canonique associée à un fibré principal,, Bulletin de la S. M. F., 113 (1985), 259.   Google Scholar

[13]

K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, volume 124 of London Mathematical Society Lecture Note Series., Cambridge University Press, (1987).  doi: 10.1017/CBO9780511661839.  Google Scholar

[14]

I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions,, Amer. J. Math., 124 (2002), 567.  doi: 10.1353/ajm.2002.0019.  Google Scholar

[15]

S. Morrison, Classifying Spinor Structures,, Master's thesis, (2001).   Google Scholar

[16]

K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids,, Topology, 39 (2000), 445.  doi: 10.1016/S0040-9383(98)00069-X.  Google Scholar

[17]

A. Nijenhuis, Theory of the Geometric Object, 1952., Doctoral thesis, ().   Google Scholar

[18]

A. Nijenhuis, Geometric aspects of formal differential operations on tensors fields,, In Proc. Internat. Congress Math. 1958, (1958), 463.   Google Scholar

[19]

A. Nijenhuis, Natural bundles and their general properties. Geometric objects revisited,, In Differential geometry (in honor of Kentaro Yano), (1972), 317.   Google Scholar

[20]

R. S. Palais and C. L. Terng, Natural bundles have finite order,, Topology, 19 (1977), 271.   Google Scholar

[21]

J. Peetre, Réctification (sic) à l'article «une caractérisation abstraite des opérateurs différentiels»,, Math. Scand., 8 (1960), 116.   Google Scholar

[22]

J. Pradines, Théorie de Lie pour les groupoides différentiables, relation entre propriétés locales et globales,, Comptes Rendus Acad. Sci. Paris A, 263 (1966), 907.   Google Scholar

[23]

S. E. Salvioli, On the theory of geometric objects,, J. Diff. Geom., 7 (1972), 257.   Google Scholar

[24]

J. A. Schouten and J. Haantjes, On the Theory of the Geometric Object,, Proc. London Math. Soc., S2-42 (1937), 2.  doi: 10.1112/plms/s2-42.1.356.  Google Scholar

[25]

M. E. Shanks and L. E. Pursell, The Lie algebra of a smooth manifold,, Proc. Amer. Math. Soc., 5 (1954), 468.  doi: 10.1090/S0002-9939-1954-0064764-3.  Google Scholar

[26]

Kōji Shiga and Toru Tsujishita, Differential representations of vector fields,, Kōdai Math. Sem. Rep., 28 (): 214.   Google Scholar

[27]

F. Takens, Derivations of vector fields,, Comp. Math., 26 (1973), 151.   Google Scholar

[28]

C. L. Terng, Natural vector bundles and natural differential operators,, Am. J. Math., 100 (1978), 775.  doi: 10.2307/2373910.  Google Scholar

[29]

A. Wundheiler, Objekte, Invarianten und Klassifikation der Geometrie,, Abh. Sem. Vektor Tenzoranal. Moskau, 4 (1937), 366.   Google Scholar

show all references

References:
[1]

A. Banyaga, The Structure of Classical Diffeomorphism Groups, volume 400 of Mathematics and its Applications,, Kluwer Academic Publishers Group, (1997).  doi: 10.1007/978-1-4757-6800-8.  Google Scholar

[2]

D. W. Barnes, Nilpotency of Lie algebras,, Math. Zeitschr., 79 (1962), 237.  doi: 10.1007/BF01193118.  Google Scholar

[3]

M. Berg, C. DeWitt-Morette, S. Gwo and E. Kramer, The pin groups in physics: C, P and T,, Rev. Math. Phys., 13 (2001), 953.  doi: 10.1142/S0129055X01000922.  Google Scholar

[4]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets,, Ann. of Math., 157 (2003), 575.  doi: 10.4007/annals.2003.157.575.  Google Scholar

[5]

D. B. A. Epstein and W. P. Thurston, Transformation groups and natural bundles,, Proc. London Math. Soc., 38 (1979), 219.  doi: 10.1112/plms/s3-38.2.219.  Google Scholar

[6]

D. B. Fuks, Cohomology of Infinite Dimensional Lie Algebras,, Contemporary Soviet Mathematics, (1986).   Google Scholar

[7]

H. Glöckner, Differentiable mappings between spaces of sections, 2002., arXiv:1308.1172., ().   Google Scholar

[8]

J. Grabowski, A. Kotov and N. Poncin, Geometric structures encoded in the Lie structure of an Atiyah algebroid,, Transform. Groups, 16 (2011), 137.  doi: 10.1007/s00031-011-9126-9.  Google Scholar

[9]

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory,, Graduate Texts in Mathematics, (1972).   Google Scholar

[10]

A. W. Knapp, Lie Groups Beyond an Introduction,, Birkhäuser, (1996).  doi: 10.1007/978-1-4757-2453-0.  Google Scholar

[11]

H. B. Lawson and M.-L. Michelsohn, Spin geometry,, Princeton University Press, (1994).   Google Scholar

[12]

P. B. A. Lecomte, Sur la suite exacte canonique associée à un fibré principal,, Bulletin de la S. M. F., 113 (1985), 259.   Google Scholar

[13]

K. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry, volume 124 of London Mathematical Society Lecture Note Series., Cambridge University Press, (1987).  doi: 10.1017/CBO9780511661839.  Google Scholar

[14]

I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions,, Amer. J. Math., 124 (2002), 567.  doi: 10.1353/ajm.2002.0019.  Google Scholar

[15]

S. Morrison, Classifying Spinor Structures,, Master's thesis, (2001).   Google Scholar

[16]

K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids,, Topology, 39 (2000), 445.  doi: 10.1016/S0040-9383(98)00069-X.  Google Scholar

[17]

A. Nijenhuis, Theory of the Geometric Object, 1952., Doctoral thesis, ().   Google Scholar

[18]

A. Nijenhuis, Geometric aspects of formal differential operations on tensors fields,, In Proc. Internat. Congress Math. 1958, (1958), 463.   Google Scholar

[19]

A. Nijenhuis, Natural bundles and their general properties. Geometric objects revisited,, In Differential geometry (in honor of Kentaro Yano), (1972), 317.   Google Scholar

[20]

R. S. Palais and C. L. Terng, Natural bundles have finite order,, Topology, 19 (1977), 271.   Google Scholar

[21]

J. Peetre, Réctification (sic) à l'article «une caractérisation abstraite des opérateurs différentiels»,, Math. Scand., 8 (1960), 116.   Google Scholar

[22]

J. Pradines, Théorie de Lie pour les groupoides différentiables, relation entre propriétés locales et globales,, Comptes Rendus Acad. Sci. Paris A, 263 (1966), 907.   Google Scholar

[23]

S. E. Salvioli, On the theory of geometric objects,, J. Diff. Geom., 7 (1972), 257.   Google Scholar

[24]

J. A. Schouten and J. Haantjes, On the Theory of the Geometric Object,, Proc. London Math. Soc., S2-42 (1937), 2.  doi: 10.1112/plms/s2-42.1.356.  Google Scholar

[25]

M. E. Shanks and L. E. Pursell, The Lie algebra of a smooth manifold,, Proc. Amer. Math. Soc., 5 (1954), 468.  doi: 10.1090/S0002-9939-1954-0064764-3.  Google Scholar

[26]

Kōji Shiga and Toru Tsujishita, Differential representations of vector fields,, Kōdai Math. Sem. Rep., 28 (): 214.   Google Scholar

[27]

F. Takens, Derivations of vector fields,, Comp. Math., 26 (1973), 151.   Google Scholar

[28]

C. L. Terng, Natural vector bundles and natural differential operators,, Am. J. Math., 100 (1978), 775.  doi: 10.2307/2373910.  Google Scholar

[29]

A. Wundheiler, Objekte, Invarianten und Klassifikation der Geometrie,, Abh. Sem. Vektor Tenzoranal. Moskau, 4 (1937), 366.   Google Scholar

[1]

Hongliang Chang, Yin Chen, Runxuan Zhang. A generalization on derivations of Lie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2020124

[2]

Ville Salo, Ilkka Törmä. Recoding Lie algebraic subshifts. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 1005-1021. doi: 10.3934/dcds.2020307

[3]

Jianfeng Huang, Haihua Liang. Limit cycles of planar system defined by the sum of two quasi-homogeneous vector fields. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 861-873. doi: 10.3934/dcdsb.2020145

[4]

Wenjun Liu, Yukun Xiao, Xiaoqing Yue. Classification of finite irreducible conformal modules over Lie conformal algebra $ \mathcal{W}(a, b, r) $. Electronic Research Archive, , () : -. doi: 10.3934/era.2020123

[5]

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

[6]

Yongjie Wang, Nan Gao. Some properties for almost cellular algebras. Electronic Research Archive, 2021, 29 (1) : 1681-1689. doi: 10.3934/era.2020086

[7]

Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128

[8]

Shudi Yang, Xiangli Kong, Xueying Shi. Complete weight enumerators of a class of linear codes over finite fields. Advances in Mathematics of Communications, 2021, 15 (1) : 99-112. doi: 10.3934/amc.2020045

[9]

Manxue You, Shengjie Li. Perturbation of Image and conjugate duality for vector optimization. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020176

[10]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[11]

Qiao Liu. Local rigidity of certain solvable group actions on tori. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 553-567. doi: 10.3934/dcds.2020269

[12]

Kien Trung Nguyen, Vo Nguyen Minh Hieu, Van Huy Pham. Inverse group 1-median problem on trees. Journal of Industrial & Management Optimization, 2021, 17 (1) : 221-232. doi: 10.3934/jimo.2019108

[13]

Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1347-1357. doi: 10.3934/dcds.2020320

[14]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164

[15]

Thomas Frenzel, Matthias Liero. Effective diffusion in thin structures via generalized gradient systems and EDP-convergence. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 395-425. doi: 10.3934/dcdss.2020345

[16]

Ying Lin, Qi Ye. Support vector machine classifiers by non-Euclidean margins. Mathematical Foundations of Computing, 2020, 3 (4) : 279-300. doi: 10.3934/mfc.2020018

[17]

Wen Li, Wei-Hui Liu, Seak Weng Vong. Perron vector analysis for irreducible nonnegative tensors and its applications. Journal of Industrial & Management Optimization, 2021, 17 (1) : 29-50. doi: 10.3934/jimo.2019097

[18]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[19]

Bing Sun, Liangyun Chen, Yan Cao. On the universal $ \alpha $-central extensions of the semi-direct product of Hom-preLie algebras. Electronic Research Archive, , () : -. doi: 10.3934/era.2021004

[20]

Kengo Matsumoto. $ C^* $-algebras associated with asymptotic equivalence relations defined by hyperbolic toral automorphisms. Electronic Research Archive, , () : -. doi: 10.3934/era.2021006

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (81)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]