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

Infinitesimally natural principal bundles

Bas Janssens 1,

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.
Keywords: diffeomorphism group, Lie algebras of vector fields., integration of Lie algebroids, spin structures, Natural bundles.
Mathematics Subject Classification: Primary: 53C27, 57R50, 22E65, 57R2.
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, Dordrecht, 1997. doi: 10.1007/978-1-4757-6800-8.  Google Scholar

[2]

D. W. Barnes, Nilpotency of Lie algebras, Math. Zeitschr., 79 (1962), 237-238. 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), 953-1034. doi: 10.1142/S0129055X01000922.  Google Scholar

[4]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620. 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), 219-236. doi: 10.1112/plms/s3-38.2.219.  Google Scholar

[6]

D. B. Fuks, Cohomology of Infinite Dimensional Lie Algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 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), 137-160. doi: 10.1007/s00031-011-9126-9.  Google Scholar

[9]

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1972.  Google Scholar

[10]

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

[11]

H. B. Lawson and M.-L. Michelsohn, Spin geometry, Princeton University Press, second edition, 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), 259-271.  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, Cambridge, 1987. doi: 10.1017/CBO9780511661839.  Google Scholar

[14]

I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math., 124 (2002), 567-593. doi: 10.1353/ajm.2002.0019.  Google Scholar

[15]

S. Morrison, Classifying Spinor Structures, Master's thesis, University of New South Wales, 2001. Google Scholar

[16]

K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids, Topology, 39 (2000), 445-467. 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, pages 463-469. Cambridge Univ. Press, New York, 1960.  Google Scholar

[19]

A. Nijenhuis, Natural bundles and their general properties. Geometric objects revisited, In Differential geometry (in honor of Kentaro Yano), pages 317-334. Kinokuniya, Tokyo, 1972.  Google Scholar

[20]

R. S. Palais and C. L. Terng, Natural bundles have finite order, Topology, 19 (1977), 271-277.  Google Scholar

[21]

J. Peetre, Réctification (sic) à l'article «une caractérisation abstraite des opérateurs différentiels», Math. Scand., 8 (1960), 116-120.  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), 907-910.  Google Scholar

[23]

S. E. Salvioli, On the theory of geometric objects, J. Diff. Geom., 7 (1972), 257-278.  Google Scholar

[24]

J. A. Schouten and J. Haantjes, On the Theory of the Geometric Object, Proc. London Math. Soc., S2-42 (1937), 356-376. 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), 468-472. 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), 151-158.  Google Scholar

[28]

C. L. Terng, Natural vector bundles and natural differential operators, Am. J. Math., 100 (1978), 775-828. doi: 10.2307/2373910.  Google Scholar

[29]

A. Wundheiler, Objekte, Invarianten und Klassifikation der Geometrie, Abh. Sem. Vektor Tenzoranal. Moskau, 4 (1937), 366-375. 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, Dordrecht, 1997. doi: 10.1007/978-1-4757-6800-8.  Google Scholar

[2]

D. W. Barnes, Nilpotency of Lie algebras, Math. Zeitschr., 79 (1962), 237-238. 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), 953-1034. doi: 10.1142/S0129055X01000922.  Google Scholar

[4]

M. Crainic and R. L. Fernandes, Integrability of Lie brackets, Ann. of Math., 157 (2003), 575-620. 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), 219-236. doi: 10.1112/plms/s3-38.2.219.  Google Scholar

[6]

D. B. Fuks, Cohomology of Infinite Dimensional Lie Algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 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), 137-160. doi: 10.1007/s00031-011-9126-9.  Google Scholar

[9]

J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1972.  Google Scholar

[10]

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

[11]

H. B. Lawson and M.-L. Michelsohn, Spin geometry, Princeton University Press, second edition, 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), 259-271.  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, Cambridge, 1987. doi: 10.1017/CBO9780511661839.  Google Scholar

[14]

I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math., 124 (2002), 567-593. doi: 10.1353/ajm.2002.0019.  Google Scholar

[15]

S. Morrison, Classifying Spinor Structures, Master's thesis, University of New South Wales, 2001. Google Scholar

[16]

K. C. H. Mackenzie and P. Xu, Integration of Lie bialgebroids, Topology, 39 (2000), 445-467. 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, pages 463-469. Cambridge Univ. Press, New York, 1960.  Google Scholar

[19]

A. Nijenhuis, Natural bundles and their general properties. Geometric objects revisited, In Differential geometry (in honor of Kentaro Yano), pages 317-334. Kinokuniya, Tokyo, 1972.  Google Scholar

[20]

R. S. Palais and C. L. Terng, Natural bundles have finite order, Topology, 19 (1977), 271-277.  Google Scholar

[21]

J. Peetre, Réctification (sic) à l'article «une caractérisation abstraite des opérateurs différentiels», Math. Scand., 8 (1960), 116-120.  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), 907-910.  Google Scholar

[23]

S. E. Salvioli, On the theory of geometric objects, J. Diff. Geom., 7 (1972), 257-278.  Google Scholar

[24]

J. A. Schouten and J. Haantjes, On the Theory of the Geometric Object, Proc. London Math. Soc., S2-42 (1937), 356-376. 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), 468-472. 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), 151-158.  Google Scholar

[28]

C. L. Terng, Natural vector bundles and natural differential operators, Am. J. Math., 100 (1978), 775-828. doi: 10.2307/2373910.  Google Scholar

[29]

A. Wundheiler, Objekte, Invarianten und Klassifikation der Geometrie, Abh. Sem. Vektor Tenzoranal. Moskau, 4 (1937), 366-375. Google Scholar

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