On Lie semiheaps and ternary principal bundles

Andrew James Bruce

Archivum Mathematicum (2024)

  • Volume: 060, Issue: 2, page 101-124
  • ISSN: 0044-8753

Abstract

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We introduce the notion of a Lie semiheap as a smooth manifold equipped with a para-associative ternary product. For a particular class of Lie semiheaps we establish the existence of left-invariant vector fields. Furthermore, we show how such manifolds are related to Lie groups and establish the analogue of principal bundles in this ternary setting. In particular, we generalise the well-known ‘heapification’ functor to the ambience of Lie groups and principal bundles.

How to cite

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Bruce, Andrew James. "On Lie semiheaps and ternary principal bundles." Archivum Mathematicum 060.2 (2024): 101-124. <http://eudml.org/doc/299263>.

@article{Bruce2024,
abstract = {We introduce the notion of a Lie semiheap as a smooth manifold equipped with a para-associative ternary product. For a particular class of Lie semiheaps we establish the existence of left-invariant vector fields. Furthermore, we show how such manifolds are related to Lie groups and establish the analogue of principal bundles in this ternary setting. In particular, we generalise the well-known ‘heapification’ functor to the ambience of Lie groups and principal bundles.},
author = {Bruce, Andrew James},
journal = {Archivum Mathematicum},
keywords = {heaps; semiheaps; principal bundles; group actions; generalised associativity},
language = {eng},
number = {2},
pages = {101-124},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {On Lie semiheaps and ternary principal bundles},
url = {http://eudml.org/doc/299263},
volume = {060},
year = {2024},
}

TY - JOUR
AU - Bruce, Andrew James
TI - On Lie semiheaps and ternary principal bundles
JO - Archivum Mathematicum
PY - 2024
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 060
IS - 2
SP - 101
EP - 124
AB - We introduce the notion of a Lie semiheap as a smooth manifold equipped with a para-associative ternary product. For a particular class of Lie semiheaps we establish the existence of left-invariant vector fields. Furthermore, we show how such manifolds are related to Lie groups and establish the analogue of principal bundles in this ternary setting. In particular, we generalise the well-known ‘heapification’ functor to the ambience of Lie groups and principal bundles.
LA - eng
KW - heaps; semiheaps; principal bundles; group actions; generalised associativity
UR - http://eudml.org/doc/299263
ER -

References

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  1. Baer, R., Zur Einführung des Scharbegriffs, J. Reine Angew. Math. 160 (1929), 199–207. (1929) MR1581184
  2. Borowiec, A., Dudek, W.A., Duplij, S., Basic concepts of ternary Hopf algebras, J. Kharkov National Univ., ser. Nuclei, Particles and Fields, vol. 529 3 (15) (2001), 21–29. (2001) 
  3. Breaz, S., Brzeziński, T., Rybołowicz, B., Saracco, P., 10.1007/s10231-023-01369-0, Ann. Mat. Pur. Appl. 203 (2024), 403–405. (2024) MR4685735DOI10.1007/s10231-023-01369-0
  4. Bruce, A.J., 10.3390/universe8010056, Universe 8 (1) (2022), 56. (2022) DOI10.3390/universe8010056
  5. Brzeziński, T., Towards semi-trusses, Rev. Roumaine Math. Pures Appl. 63 (2) (2018), 75–89. (2018) MR3812011
  6. Brzeziński, T., 10.1090/tran/7705, Trans. Amer. Math. Soc. 372 (6) (2019), 4149–4176. (2019) MR4009388DOI10.1090/tran/7705
  7. Brzeziński, T., 10.1016/j.jpaa.2019.106258, J. Pure Appl. Algebra 224 (6) (2020), 39 pp., 106258. (2020) MR4048518DOI10.1016/j.jpaa.2019.106258
  8. Brzeziński, T., Lie trusses and heaps of Lie affebras, PoS 406 (2022), 12 pp., 307. (2022) 
  9. Brzeziński, T., 10.1017/S0013091523000275, Proc. Edinb. Math. Soc. 66 (2) (2023), 548–556. (2023) MR4618086DOI10.1017/S0013091523000275
  10. Brzeziński, T., Rybołowicz, B., 10.1007/s10468-020-10008-8, Algebr. Represent. Theory 25 (1) (2022), 1–23. (2022) MR4368576DOI10.1007/s10468-020-10008-8
  11. Brzeziński, T., Wisbauer, R., Corings and comodules, London Math. Soc. Lecture Note Ser., vol. 309, Cambridge University Press, Cambridge, 2003, pp. xii+476 pp. (2003) MR2012570
  12. Grabowska, K., Grabowski, J., Urbański, P., 10.1023/A:1024457728027, Ann. Global Anal.Geom. 24 (2003), 101–130. (2003) MR1990111DOI10.1023/A:1024457728027
  13. Grabowski, J., An introduction to loopoids, Comment. Math. Univ. Carolin. 57 (2016), 515–526. (2016) MR3583303
  14. Grabowski, J., Z., Ravanpak, 10.1016/j.difgeo.2022.101887, Differential Geom. Appl. 82 (2022), 32 pp., Paper No. 101887. (2022) MR4404504DOI10.1016/j.difgeo.2022.101887
  15. Hollings, C.D., Lawson, M.V., Wagner’s theory of generalised heaps, Springer, Cham, 2017, xv+189 pp., ISBN: 978-3-319-63620-7. (2017) MR3729305
  16. Kerner, R., 10.1142/S0219887808003326, Int. J. Geom. Methods Mod. Phys. 5 (2008), 1265–1294. (2008) MR2484553DOI10.1142/S0219887808003326
  17. Kerner, R., Ternary generalizations of graded algebras with some physical applications, Rev. Roumaine Math. Pures Appl 63 (2018), 107–141. (2018) MR3812013
  18. Kolář, I., Michor, P.W., Slovák, J., Natural operations in differential geometry, Springer-Verlag, Berlin, 1993. (1993) Zbl0782.53013MR1202431
  19. Konstantinova, L.I., Semiheap bundles, Saratov Gos. Univ. Saratov No. 4 (1978), 46–54. (1978) MR0538067
  20. Kontsevich, M., 10.1023/A:1007555725247, Lett. Math. Phys. 48 (1999), 35–72. (1999) MR1718044DOI10.1023/A:1007555725247
  21. Kosmann-Schwarzbach, Y., Multiplicativity, from Lie groups to generalized geometry, Banach Center Publ. 110 (2016), 131–166. (2016) MR3642395
  22. Mac Lane, S., Categories for the working mathematician, Grad. Texts in Math., vol. 5, New York, NY: Springer, 2nd ed., 1998, pp. xii+314 pp. (1998) MR1712872
  23. Nakahara, M., Geometry, topology and physics, 2nd ed., Graduate Stud. Ser. Phys. Bristol: Institute of Physics (IOP), 2003, xxii+ 573 pp. (2003) MR2001829
  24. Prüfer, H., Theorie der Abelschen Gruppen, Math. Z. 20 (1) (1924), 165–171. (1924) MR1544670
  25. Saito, M., E., Zappala, Braided Frobenius algebras from certain Hopf algebras, J. Algebra Appl. 22 (1) (2023), 23 pp., Paper No. 2350012. (2023) MR4526173
  26. Škoda, Z., Quantum heaps, cops and heapy categories, Math. Commun. 12 (1) (2007), 1–9. (2007) MR2420028
  27. Trèves, F., Topological Vector Spaces, Distributions and Kernels, Dover Publications, Inc., Mineola, NY, 2006, xvi+565 pp., ISBN: 0-486-45352-9. (2006) MR2296978

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