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Modular class of Lie $ \infty $-algebroids and adjoint representations
Quotients of double vector bundles and multigraded bundles
Mathematics Department, University of Toronto, 40 St George Street, Toronto, ON M5S 2E4, Canada |
We study quotients of multi-graded bundles, including double vector bundles. Among other things, we show that any such quotient fits into a tower of affine bundles. Applications of the theory include a construction of normal bundles for weighted submanifolds, as well as for pairs of submanifolds with clean intersection.
References:
[1] |
Z. Chen, Z. J. Liu and Y. H. Sheng,
On double vector bundles, Acta Math. Sin. (Engl. Ser.), 30 (2014), 1655-1673.
doi: 10.1007/s10114-014-2412-4. |
[2] |
F. del Carpio-Marek, Geometric Structures on Degree $2$ Manifolds, Ph.D. thesis, IMPA, 2015. |
[3] |
C. Ehresmann, Prolongements des catégories différentiables, in Topologie et Géométrie Différentielle (Séminaire Ehresmann, Vol. VI, 1964), Inst. Henri Poincaré, Paris, 1964, 8pp. |
[4] |
V. Fischer and M. Ruzhansky, Quantization on Nilpotent Lie Groups, Progress in Mathematics, 314, Birkhäuser/Springer, Cham, 2016.
doi: 10.1007/978-3-319-29558-9. |
[5] |
M. K. Flari and K. Mackenzie,
Warps, grids and curvature in triple vector bundles, Lett. Math. Phys., 109 (2019), 135-185.
doi: 10.1007/s11005-018-1103-y. |
[6] |
J. Grabowski and M. Rotkiewicz,
Graded bundles and homogeneity structures, J. Geom. Phys., 62 (2012), 21-36.
doi: 10.1016/j.geomphys.2011.09.004. |
[7] |
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. |
[8] |
A. Gracia-Saz and K. C. H. Mackenzie, Duality functors for $n$-fold vector bundles, preprint, arXiv: 1209.0027. |
[9] |
A. Gracia-Saz and K. C. H. Mackenzie,
Duality functors for triple vector bundles, Lett. Math. Phys., 90 (2009), 175-200.
doi: 10.1007/s11005-009-0346-z. |
[10] |
A. Gracia-Saz and R. A. Mehta,
Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Adv. Math., 223 (2010), 1236-1275.
doi: 10.1016/j.aim.2009.09.010. |
[11] |
A. Gracia-Saz and R. A. Mehta,
$\mathcal VB$-groupoids and representation theory of Lie groupoids, J. Symplectic Geom., 15 (2017), 741-783.
doi: 10.4310/JSG.2017.v15.n3.a5. |
[12] |
M. Heuer and M. Jotz Lean, Multiple vector bundles: Cores, splittings and decompositions, Theory Appl. Categ., 35 (2020), 665–699. Available from: http://www.tac.mta.ca/tac/volumes/35/19/35-19.pdf. |
[13] |
I. Kolář, P. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-662-02950-3. |
[14] |
Y. Loizides and E. Meinrenken, Differential geometry of weightings, preprint, arXiv: 2010.01643. |
[15] |
K. C. H. Mackenzie,
Double Lie algebroids and second-order geometry. I., Adv. Math., 94 (1992), 180-239.
doi: 10.1016/0001-8708(92)90036-K. |
[16] |
K. C. H. Mackenzie, Duality and triple structures, in The Breadth of Symplectic and Poisson Geometry, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005,455–481.
doi: 10.1007/0-8176-4419-9_15. |
[17] |
K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9781107325883. |
[18] |
K. C. H. Mackenzie and P. Xu,
Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.
doi: 10.1215/S0012-7094-94-07318-3. |
[19] |
E. Meinrenken,
Euler-like vector fields, normal forms, and isotropic embeddings, Indag. Math. (N.S.), 32 (2021), 224-245.
doi: 10.1016/j.indag.2020.08.006. |
[20] |
E. Meinrenken and J. Pike,
The Weil algebra for double Lie algebroids, Int. Math. Res. Not. IMRN, 2021 (2021), 8550-8622.
doi: 10.1093/imrn/rnz361. |
[21] |
R. B. Melrose, Differential Analysis on Manifolds with Corners, manuscript. Available from: http://www-math.mit.edu/ rbm/book.html. |
[22] |
A. Morimoto,
Liftings of tensor fields and connections to tangent bundles of higher order, Nagoya Math. J., 40 (1970), 99-120.
doi: 10.1017/S002776300001388X. |
[23] |
T. Nagano,
$1$-forms with the exterior derivative of maximal rank, J. Differential Geometry, 2 (1968), 253-264.
doi: 10.4310/jdg/1214428439. |
[24] |
J. Pike, Weil Algebras and Double Lie Algebroids, Ph.D. thesis, University of Toronto, 2020. Available from: https://tspace.library.utoronto.ca/bitstream/1807/103377/4/Pike_Jeffrey_202011_PhD_thesis.pdf. |
[25] |
J. Pradines, Fibres Vectoriels Doubles et Calcul des Jets non Holonomes, Esquisses Mathématiques, 29, Université d'Amiens, U.E.R. de Mathématiques, Amiens, 1977. |
[26] |
J. Pradines,
Représentation des jets non holonomes par des morphismes vectoriels doubles soudés, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 1523-1526.
|
show all references
References:
[1] |
Z. Chen, Z. J. Liu and Y. H. Sheng,
On double vector bundles, Acta Math. Sin. (Engl. Ser.), 30 (2014), 1655-1673.
doi: 10.1007/s10114-014-2412-4. |
[2] |
F. del Carpio-Marek, Geometric Structures on Degree $2$ Manifolds, Ph.D. thesis, IMPA, 2015. |
[3] |
C. Ehresmann, Prolongements des catégories différentiables, in Topologie et Géométrie Différentielle (Séminaire Ehresmann, Vol. VI, 1964), Inst. Henri Poincaré, Paris, 1964, 8pp. |
[4] |
V. Fischer and M. Ruzhansky, Quantization on Nilpotent Lie Groups, Progress in Mathematics, 314, Birkhäuser/Springer, Cham, 2016.
doi: 10.1007/978-3-319-29558-9. |
[5] |
M. K. Flari and K. Mackenzie,
Warps, grids and curvature in triple vector bundles, Lett. Math. Phys., 109 (2019), 135-185.
doi: 10.1007/s11005-018-1103-y. |
[6] |
J. Grabowski and M. Rotkiewicz,
Graded bundles and homogeneity structures, J. Geom. Phys., 62 (2012), 21-36.
doi: 10.1016/j.geomphys.2011.09.004. |
[7] |
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. |
[8] |
A. Gracia-Saz and K. C. H. Mackenzie, Duality functors for $n$-fold vector bundles, preprint, arXiv: 1209.0027. |
[9] |
A. Gracia-Saz and K. C. H. Mackenzie,
Duality functors for triple vector bundles, Lett. Math. Phys., 90 (2009), 175-200.
doi: 10.1007/s11005-009-0346-z. |
[10] |
A. Gracia-Saz and R. A. Mehta,
Lie algebroid structures on double vector bundles and representation theory of Lie algebroids, Adv. Math., 223 (2010), 1236-1275.
doi: 10.1016/j.aim.2009.09.010. |
[11] |
A. Gracia-Saz and R. A. Mehta,
$\mathcal VB$-groupoids and representation theory of Lie groupoids, J. Symplectic Geom., 15 (2017), 741-783.
doi: 10.4310/JSG.2017.v15.n3.a5. |
[12] |
M. Heuer and M. Jotz Lean, Multiple vector bundles: Cores, splittings and decompositions, Theory Appl. Categ., 35 (2020), 665–699. Available from: http://www.tac.mta.ca/tac/volumes/35/19/35-19.pdf. |
[13] |
I. Kolář, P. Michor and J. Slovák, Natural Operations in Differential Geometry, Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-662-02950-3. |
[14] |
Y. Loizides and E. Meinrenken, Differential geometry of weightings, preprint, arXiv: 2010.01643. |
[15] |
K. C. H. Mackenzie,
Double Lie algebroids and second-order geometry. I., Adv. Math., 94 (1992), 180-239.
doi: 10.1016/0001-8708(92)90036-K. |
[16] |
K. C. H. Mackenzie, Duality and triple structures, in The Breadth of Symplectic and Poisson Geometry, Progr. Math., 232, Birkhäuser Boston, Boston, MA, 2005,455–481.
doi: 10.1007/0-8176-4419-9_15. |
[17] |
K. C. H. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids, London Mathematical Society Lecture Note Series, 213, Cambridge University Press, Cambridge, 2005.
doi: 10.1017/CBO9781107325883. |
[18] |
K. C. H. Mackenzie and P. Xu,
Lie bialgebroids and Poisson groupoids, Duke Math. J., 73 (1994), 415-452.
doi: 10.1215/S0012-7094-94-07318-3. |
[19] |
E. Meinrenken,
Euler-like vector fields, normal forms, and isotropic embeddings, Indag. Math. (N.S.), 32 (2021), 224-245.
doi: 10.1016/j.indag.2020.08.006. |
[20] |
E. Meinrenken and J. Pike,
The Weil algebra for double Lie algebroids, Int. Math. Res. Not. IMRN, 2021 (2021), 8550-8622.
doi: 10.1093/imrn/rnz361. |
[21] |
R. B. Melrose, Differential Analysis on Manifolds with Corners, manuscript. Available from: http://www-math.mit.edu/ rbm/book.html. |
[22] |
A. Morimoto,
Liftings of tensor fields and connections to tangent bundles of higher order, Nagoya Math. J., 40 (1970), 99-120.
doi: 10.1017/S002776300001388X. |
[23] |
T. Nagano,
$1$-forms with the exterior derivative of maximal rank, J. Differential Geometry, 2 (1968), 253-264.
doi: 10.4310/jdg/1214428439. |
[24] |
J. Pike, Weil Algebras and Double Lie Algebroids, Ph.D. thesis, University of Toronto, 2020. Available from: https://tspace.library.utoronto.ca/bitstream/1807/103377/4/Pike_Jeffrey_202011_PhD_thesis.pdf. |
[25] |
J. Pradines, Fibres Vectoriels Doubles et Calcul des Jets non Holonomes, Esquisses Mathématiques, 29, Université d'Amiens, U.E.R. de Mathématiques, Amiens, 1977. |
[26] |
J. Pradines,
Représentation des jets non holonomes par des morphismes vectoriels doubles soudés, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 1523-1526.
|
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