Infinitesimally natural principal bundles

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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

Abstract
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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.
[2]	D. W. Barnes, Nilpotency of Lie algebras, Math. Zeitschr., 79 (1962), 237-238. doi: 10.1007/BF01193118.
[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.
[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.
[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.
[6]	D. B. Fuks, Cohomology of Infinite Dimensional Lie Algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986.
[7]	H. Glöckner, Differentiable mappings between spaces of sections, 2002. arXiv:1308.1172.
[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.
[9]	J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1972.
[10]	A. W. Knapp, Lie Groups Beyond an Introduction, Birkhäuser, Boston, first edition, 1996. doi: 10.1007/978-1-4757-2453-0.
[11]	H. B. Lawson and M.-L. Michelsohn, Spin geometry, Princeton University Press, second edition, 1994.
[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.
[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.
[14]	I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math., 124 (2002), 567-593. doi: 10.1353/ajm.2002.0019.
[15]	S. Morrison, Classifying Spinor Structures, Master's thesis, University of New South Wales, 2001.
[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.
[17]	A. Nijenhuis, Theory of the Geometric Object, 1952. Doctoral thesis, Universiteit van Amsterdam.
[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.
[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.
[20]	R. S. Palais and C. L. Terng, Natural bundles have finite order, Topology, 19 (1977), 271-277.
[21]	J. Peetre, Réctification (sic) à l'article «une caractérisation abstraite des opérateurs différentiels», Math. Scand., 8 (1960), 116-120.
[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.
[23]	S. E. Salvioli, On the theory of geometric objects, J. Diff. Geom., 7 (1972), 257-278.
[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.
[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.
[26]	Kōji Shiga and Toru Tsujishita, Differential representations of vector fields, Kōdai Math. Sem. Rep., 28 (1976/77), 214-225.
[27]	F. Takens, Derivations of vector fields, Comp. Math., 26 (1973), 151-158.
[28]	C. L. Terng, Natural vector bundles and natural differential operators, Am. J. Math., 100 (1978), 775-828. doi: 10.2307/2373910.
[29]	A. Wundheiler, Objekte, Invarianten und Klassifikation der Geometrie, Abh. Sem. Vektor Tenzoranal. Moskau, 4 (1937), 366-375.

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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.
[2]	D. W. Barnes, Nilpotency of Lie algebras, Math. Zeitschr., 79 (1962), 237-238. doi: 10.1007/BF01193118.
[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.
[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.
[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.
[6]	D. B. Fuks, Cohomology of Infinite Dimensional Lie Algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986.
[7]	H. Glöckner, Differentiable mappings between spaces of sections, 2002. arXiv:1308.1172.
[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.
[9]	J. E. Humphreys, Introduction to Lie Algebras and Representation Theory, Graduate Texts in Mathematics, Springer-Verlag, New York, 1972.
[10]	A. W. Knapp, Lie Groups Beyond an Introduction, Birkhäuser, Boston, first edition, 1996. doi: 10.1007/978-1-4757-2453-0.
[11]	H. B. Lawson and M.-L. Michelsohn, Spin geometry, Princeton University Press, second edition, 1994.
[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.
[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.
[14]	I. Moerdijk and J. Mrčun, On integrability of infinitesimal actions, Amer. J. Math., 124 (2002), 567-593. doi: 10.1353/ajm.2002.0019.
[15]	S. Morrison, Classifying Spinor Structures, Master's thesis, University of New South Wales, 2001.
[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.
[17]	A. Nijenhuis, Theory of the Geometric Object, 1952. Doctoral thesis, Universiteit van Amsterdam.
[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.
[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.
[20]	R. S. Palais and C. L. Terng, Natural bundles have finite order, Topology, 19 (1977), 271-277.
[21]	J. Peetre, Réctification (sic) à l'article «une caractérisation abstraite des opérateurs différentiels», Math. Scand., 8 (1960), 116-120.
[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.
[23]	S. E. Salvioli, On the theory of geometric objects, J. Diff. Geom., 7 (1972), 257-278.
[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.
[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.
[26]	Kōji Shiga and Toru Tsujishita, Differential representations of vector fields, Kōdai Math. Sem. Rep., 28 (1976/77), 214-225.
[27]	F. Takens, Derivations of vector fields, Comp. Math., 26 (1973), 151-158.
[28]	C. L. Terng, Natural vector bundles and natural differential operators, Am. J. Math., 100 (1978), 775-828. doi: 10.2307/2373910.
[29]	A. Wundheiler, Objekte, Invarianten und Klassifikation der Geometrie, Abh. Sem. Vektor Tenzoranal. Moskau, 4 (1937), 366-375.

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