| Title:
|
On Lie semiheaps and ternary principal bundles (English) |
| Author:
|
Bruce, Andrew James |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
60 |
| Issue:
|
2 |
| Year:
|
2024 |
| Pages:
|
101-124 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
heaps |
| Keyword:
|
semiheaps |
| Keyword:
|
principal bundles |
| Keyword:
|
group actions |
| Keyword:
|
generalised associativity |
| MSC:
|
20N10 |
| MSC:
|
22E15 |
| DOI:
|
10.5817/AM2024-2-101 |
| . |
| Date available:
|
2024-04-04T12:05:15Z |
| Last updated:
|
2024-04-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152308 |
| . |
| Reference:
|
[1] Baer, R.: Zur Einführung des Scharbegriffs.J. Reine Angew. Math. 160 (1929), 199–207. MR 1581184 |
| Reference:
|
[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. |
| Reference:
|
[3] Breaz, S., Brzeziński, T., Rybołowicz, B., Saracco, P.: Heaps of modules and affine spaces.Ann. Mat. Pur. Appl. 203 (2024), 403–405. MR 4685735, 10.1007/s10231-023-01369-0 |
| Reference:
|
[4] Bruce, A.J.: Semiheaps and Ternary Algebras in Quantum Mechanics Revisited,.Universe 8 (1) (2022), 56. 10.3390/universe8010056 |
| Reference:
|
[5] Brzeziński, T.: Towards semi-trusses.Rev. Roumaine Math. Pures Appl. 63 (2) (2018), 75–89. MR 3812011 |
| Reference:
|
[6] Brzeziński, T.: Trusses: between braces and rings.Trans. Amer. Math. Soc. 372 (6) (2019), 4149–4176. MR 4009388, 10.1090/tran/7705 |
| Reference:
|
[7] Brzeziński, T.: Trusses: paragons, ideals and modules.J. Pure Appl. Algebra 224 (6) (2020), 39 pp., 106258. MR 4048518, 10.1016/j.jpaa.2019.106258 |
| Reference:
|
[8] Brzeziński, T.: Lie trusses and heaps of Lie affebras.PoS 406 (2022), 12 pp., 307. |
| Reference:
|
[9] Brzeziński, T.: The algebra of elliptic curves.Proc. Edinb. Math. Soc. 66 (2) (2023), 548–556. MR 4618086, 10.1017/S0013091523000275 |
| Reference:
|
[10] Brzeziński, T., Rybołowicz, B.: Modules over trusses vs modules over rings: direct sums and free modules.Algebr. Represent. Theory 25 (1) (2022), 1–23. MR 4368576, 10.1007/s10468-020-10008-8 |
| Reference:
|
[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. MR 2012570 |
| Reference:
|
[12] Grabowska, K., Grabowski, J., Urbański, P.: Lie brackets on affine bundles.Ann. Global Anal.Geom. 24 (2003), 101–130. MR 1990111, 10.1023/A:1024457728027 |
| Reference:
|
[13] Grabowski, J.: An introduction to loopoids.Comment. Math. Univ. Carolin. 57 (2016), 515–526. MR 3583303 |
| Reference:
|
[14] Grabowski, J., Z., Ravanpak: Nonassociative analogs of Lie groupoids.Differential Geom. Appl. 82 (2022), 32 pp., Paper No. 101887. MR 4404504, 10.1016/j.difgeo.2022.101887 |
| Reference:
|
[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. MR 3729305 |
| Reference:
|
[16] Kerner, R.: Ternary and non-associative structures.Int. J. Geom. Methods Mod. Phys. 5 (2008), 1265–1294. MR 2484553, 10.1142/S0219887808003326 |
| Reference:
|
[17] Kerner, R.: Ternary generalizations of graded algebras with some physical applications.Rev. Roumaine Math. Pures Appl 63 (2018), 107–141. MR 3812013 |
| Reference:
|
[18] Kolář, I., Michor, P.W., Slovák, J.: Natural operations in differential geometry.Springer-Verlag, Berlin, 1993. Zbl 0782.53013, MR 1202431 |
| Reference:
|
[19] Konstantinova, L.I.: Semiheap bundles.Saratov Gos. Univ. Saratov No. 4 (1978), 46–54. MR 0538067 |
| Reference:
|
[20] Kontsevich, M.: Operads and motives in deformation quantization.Lett. Math. Phys. 48 (1999), 35–72. MR 1718044, 10.1023/A:1007555725247 |
| Reference:
|
[21] Kosmann-Schwarzbach, Y.: Multiplicativity, from Lie groups to generalized geometry.Banach Center Publ. 110 (2016), 131–166. MR 3642395 |
| Reference:
|
[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. MR 1712872 |
| Reference:
|
[23] Nakahara, M.: Geometry, topology and physics.2nd ed., Graduate Stud. Ser. Phys. Bristol: Institute of Physics (IOP), 2003, xxii+ 573 pp. MR 2001829 |
| Reference:
|
[24] Prüfer, H.: Theorie der Abelschen Gruppen.Math. Z. 20 (1) (1924), 165–171. MR 1544670 |
| Reference:
|
[25] Saito, M., E., Zappala: Braided Frobenius algebras from certain Hopf algebras.J. Algebra Appl. 22 (1) (2023), 23 pp., Paper No. 2350012. MR 4526173 |
| Reference:
|
[26] Škoda, Z.: Quantum heaps, cops and heapy categories.Math. Commun. 12 (1) (2007), 1–9. MR 2420028 |
| Reference:
|
[27] Trèves, F.: Topological Vector Spaces, Distributions and Kernels.Dover Publications, Inc., Mineola, NY, 2006, xvi+565 pp., ISBN: 0-486-45352-9. MR 2296978 |
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