| Title:
|
On F-algebroids and Dubrovin’s duality (English) |
| Author:
|
Cruz Morales, John Alexander |
| Author:
|
Torres-Gomez, Alexander |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
|
0044-8753 (print) |
| ISSN:
|
1212-5059 (online) |
| Volume:
|
55 |
| Issue:
|
2 |
| Year:
|
2019 |
| Pages:
|
109-122 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this note we introduce the concept of F-algebroid, and give its elementary properties and some examples. We provide a description of the almost duality for Frobenius manifolds, introduced by Dubrovin, in terms of a composition of two anchor maps of a unique cotangent F-algebroid. (English) |
| Keyword:
|
F-manifolds |
| Keyword:
|
Frobenius manifolds |
| Keyword:
|
Lie algebroids |
| MSC:
|
53C15 |
| MSC:
|
53D45 |
| idZBL:
|
Zbl 07088762 |
| idMR:
|
MR3964438 |
| DOI:
|
10.5817/AM2019-2-109 |
| . |
| Date available:
|
2019-06-07T14:52:16Z |
| Last updated:
|
2020-02-27 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/147750 |
| . |
| Reference:
|
[1] Audin, M.: Symplectic geometry in Frobenius manifolds and quantum cohomology.J. Geom. Phys. 25 (1–2) (1998), 183–204. MR 1611969, 10.1016/S0393-0440(97)00026-0 |
| Reference:
|
[2] Crainic, M., Fernandes, R.L.: Lectures integrability Lie brackets.Geom. Topol. Monogr. 17 (2011), 1–107. MR 2795150 |
| Reference:
|
[3] David, L., Strachan, I.A.B.: Dubrovin’s duality for F-manifolds with eventual identities.Adv. Math. 226 (4) (2011), 4031–4060. MR 2770440, 10.1016/j.aim.2010.11.006 |
| Reference:
|
[4] Dotsenko, V.: Algebraic structures of F-manifolds via pre-Lie algebras.Ann. Mat. Pura Appl. (4) 198 (2019), 517–527. MR 3927168, 10.1007/s10231-018-0787-z |
| Reference:
|
[5] Dubrovin, B.: Geometry of 2D topological field theories.Lecture Notes in Math., vol. 1620, Springer, 1996. Zbl 0841.58065, MR 1397274 |
| Reference:
|
[6] Dubrovin, B.: On almost duality for Frobenius manifolds.Amer. Math. Soc. Transl. 212 (2004), 75–132. MR 2070050 |
| Reference:
|
[7] Dubrovin, B.: WDVV Equations and Frobenius Manifolds.Encyclopedia of Mathematical Physics, vol. 1, Elsevier, 2006, pp. 438–447. |
| Reference:
|
[8] Dufour, J.P., Zung, N.T.: Poisson Structures and Their Normal Forms.Birkhauser, 2000. MR 2178041 |
| Reference:
|
[9] Fernandes, R.L.: Lie algebroids, holonomy and characteristic classes.Adv. Math. 170 (1) (2002), 119–179. MR 1929305, 10.1006/aima.2001.2070 |
| Reference:
|
[10] Hertling, C.: Frobenius manifolds and moduli spaces for singularities.Cambridge University Press, 2004. MR 1924259 |
| Reference:
|
[11] Hertling, C., Manin, Y.: Weak Frobenius manifolds.Internat. Math. Res. Notices 6 (1999), 277–286. Zbl 0960.58003, MR 1680372, 10.1155/S1073792899000148 |
| Reference:
|
[12] Hitchin, N.: Frobenius manifolds.Gauge Theory and Symplectic Geometry, Springer, 1997. MR 1461570 |
| Reference:
|
[13] Kodaira, K.: Complex Manifolds and Deformation of Complex Structures.Springer, 2005. MR 2109686 |
| Reference:
|
[14] Mackenzie, K.C.H.: General Theory of Lie Groupoids and Lie Algebroids.Cambridge University Press, 2005. Zbl 1078.58011, MR 2157566 |
| Reference:
|
[15] Manetti, M.: Lectures on deformation of complex manifolds.Rendiconti di Matematica 24 (2004), 1–183. MR 2130146 |
| Reference:
|
[16] Manin, Y.: Mirrors, functoriality, and derived geometry.arXiv:1708.02849. |
| Reference:
|
[17] Manin, Y.: Frobenius Manifolds, Quantum Cohomology, and Moduli Spaces.Amer. Math. Soc. Colloq. Publ. 47 (1999), xiv+303 pp. Zbl 0952.14032, MR 1702284 |
| Reference:
|
[18] Manin, Y.: F-manifolds with flat structure and Dubrovin’s duality.Adv. Math. 198 (1) (2005), 5–26. MR 2183247, 10.1016/j.aim.2004.12.003 |
| Reference:
|
[19] Manin, Y.: Grothendieck-Verdier duality patterns in quantum algebra.Izv. Ross. Akad. Nauk Ser. Mat. 81 (4) (2017), 158–166. MR 3682786 |
| Reference:
|
[20] Weinstein, A.: Linearization problems Lie algebroids and Lie groupoids.Lett. Math. Phys. 52 (2000), 93–102. MR 1800493, 10.1023/A:1007657920231 |
| . |
