| Title:
|
$1$-cocycles on the group of contactomorphisms on the supercircle $S^{1|3}$ generalizing the Schwarzian derivative (English) |
| Author:
|
Agrebaoui, Boujemaa |
| Author:
|
Hattab, Raja |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
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66 |
| Issue:
|
4 |
| Year:
|
2016 |
| Pages:
|
1143-1163 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The relative cohomology ${\rm H}^1_{\rm diff}(\mathbb {K}(1|3),\mathfrak {osp}(2,3);{\mathcal {D}}_{\lambda ,\mu }(S^{1|3}))$ of the contact Lie superalgebra $\mathbb {K}(1|3)$ with coefficients in the space of differential operators ${\mathcal {D}}_{\lambda ,\mu }(S^{1|3})$ acting on tensor densities on $S^{1|3}$, is calculated in {N. Ben Fraj, I. Laraied, S. Omri} (2013) and the generating $1$-cocycles are expressed in terms of the infinitesimal super-Schwarzian derivative $1$-cocycle $s(X_f)=D_1D_2D_3(f)\alpha _3^{1/2}$, $X_f\in \mathbb {K}(1|3)$ which is invariant with respect to the conformal subsuperalgebra $\mathfrak {osp}(2,3)$ of $\mathbb {K}(1|3)$. \endgraf In this work we study the supergroup case. We give an explicit construction of $1$-cocycles of the group of contactomorphisms ${\mathcal {K}}(1|3)$ on the supercircle $S^{1|3}$ generating the relative cohomology ${\rm H}^1_{\rm diff}({\mathcal {K}}(1|3)$, ${\rm PC}(2,3)$; ${\mathcal {D}}_{{\lambda },\mu }(S^{1|3})$ with coefficients in ${\mathcal {D}}_{{\lambda },\mu }(S^{1|3})$. We show that they possess properties similar to those of the super-Schwarzian derivative $1$-cocycle $S_{3}(\Phi )=E_{\Phi }^{-1}(D_{1}(D_{2}),D_{3})\alpha _{3}^{1/2}$, $\Phi \in {\mathcal {K}}(1|3)$ introduced by Radul which is invariant with respect to the conformal group ${\rm PC}(2,3)$ of ${\mathcal {K}}(1|3)$. These cocycles are expressed in terms of $S_{3}(\Phi )$ and possess its properties. (English) |
| Keyword:
|
contact vector field |
| Keyword:
|
cohomology of groups |
| Keyword:
|
group of contactomorphisms |
| Keyword:
|
super-Schwarzian derivative |
| Keyword:
|
invariant differential operator |
| MSC:
|
13N10 |
| MSC:
|
17B56 |
| MSC:
|
17B66 |
| MSC:
|
20G10 |
| MSC:
|
20J06 |
| MSC:
|
53D10 |
| MSC:
|
58A50 |
| idZBL:
|
Zbl 06674867 |
| idMR:
|
MR3572928 |
| DOI:
|
10.1007/s10587-016-0315-5 |
| . |
| Date available:
|
2016-11-26T20:55:54Z |
| Last updated:
|
2020-07-03 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/145924 |
| . |
| Reference:
|
[1] Agrebaoui, B., Dammak, O., Mansour, S.: $1$-cocycle on the group of contactomorphisms on the suppercircles $S^{1|1}$ and $S^{1|2}$ generalizing the Schwarzian derivative.J. Geom. Phys. 75 (2014), 230-247. MR 3126945, 10.1016/j.geomphys.2013.10.003 |
| Reference:
|
[2] Agrebaoui, B., Mansour, S.: On the cohomology of the Lie superalgebra of contact vector fields on $S^{1| m}$.Comm. Algebra 38 (2010), 382-404. Zbl 1191.17005, MR 2597503, 10.1080/00927870903357735 |
| Reference:
|
[3] Basdouri, I., Ammar, M. Ben, Fraj, N. Ben, Boujelbane, M., Kammoun, K.: Cohomology of the Lie superalgebra of contact vector fields on $\mathbb{R}^{1|1}$ and deformations of the superspace of symbols.J. Nonlinear Math. Phys. 16 (2009), 373-409. MR 2606126, 10.1142/S1402925109000431 |
| Reference:
|
[4] Fraj, N. Ben: Cohomology of ${\cal K}(2)$ acting on linear differential operators on the superspace $\mathbb R^{1|2}$.Lett. Math. Phys. 86 (2008), 159-175. MR 2465752, 10.1007/s11005-008-0283-2 |
| Reference:
|
[5] Fraj, N. Ben, Laraied, I., Omri, S.: Supertransvectants, cohomology and deformations.J. Math. Phys. 54 (2013), 023501, 19 pages. MR 3076388 |
| Reference:
|
[6] Bernstein, J., Leites, D., Molotkov, V., Shander, V.: Seminar on Supersymmetry (v. 1. Algebra and Calcuculus: Main chapters).D. Leites Moscow Center for Continuous Mathematical Education Moskva (2011), Russian. |
| Reference:
|
[7] Bouarroudj, S.: Remarks on the Schwarzian derivatives and the invariant quantization by means of a Finsler function.Indag. Math. 15 (2004), 321-338. Zbl 1064.53060, MR 2093162, 10.1016/S0019-3577(04)80002-8 |
| Reference:
|
[8] Bouarroudj, S., Ovsienko, V.: Three cocycles on $ Diff (S^1)$ generalizing the Schwarzian derivative.Int. Math. Res. Not. 1998 (1998), 25-39. Zbl 0919.57026, MR 1601874, 10.1155/S1073792898000038 |
| Reference:
|
[9] Bouarroudj, S., Ovsienko, V.: Riemannian curl in contact geometry.Int. Math. Res. Not. 12 (2015), 3917-3942. Zbl 1330.53105, MR 3356744 |
| Reference:
|
[10] Cartan, É.: Leçons sur la Théorie des Espaces à Connexion Projective.French Paris Gauthier-Villars (Cahiers scientifiques, fasc. XVII) (1937). Zbl 0016.07603 |
| Reference:
|
[11] Conley, C. H.: Conformal symbols and the action of contact vector fields over the superline.J. Reine Angew. Math. 633 (2009), 115-163. Zbl 1248.17017, MR 2561198 |
| Reference:
|
[12] Fuks, D. B.: Cohomology of Infinite-Dimensional Lie Algebras.Contemporary Soviet Mathematics Consultants Bureau, New York (1986). Zbl 0667.17005, MR 0874337 |
| Reference:
|
[13] Gargoubi, H., Mellouli, N., Ovsienko, V.: Differential operators on supercircle: conformally equivariant quantization and symbol calculus.Lett. Math. Phys. (2007), 79 51-65. Zbl 1112.53066, MR 2290336, 10.1007/s11005-006-0129-8 |
| Reference:
|
[14] Gargoubi, H., Ovsienko, V.: Supertransvectants and symplectic geometry.Int. Math. Res. Notices 2008 Article ID rnn021, 19 pages (2008). Zbl 1144.53100, MR 2429252 |
| Reference:
|
[15] Lecomte, P. B. A., Ovsienko, V. Yu.: Projectively equivariant symbol calculus.Lett. Math. Phys. 49 (1999), 173-196. Zbl 0989.17015, MR 1743456, 10.1023/A:1007662702470 |
| Reference:
|
[16] Manin, Yu. I.: Gauge Fields and Complex Geometry.Nauka Moskva (1984), Russian. Zbl 0576.53002, MR 0787979 |
| Reference:
|
[17] Michel, J.-P., Duval, C.: On the projective geometry of the supercircle: a unified construction of the super cross-ratio and Schwarzian derivative.Int. Math. Res. Not. 2008 (2008), Article ID rnn054, 47 pages. Zbl 1145.53005, MR 2440332 |
| Reference:
|
[18] Ovsienko, V.: Lagrange Schwarzian derivative and symplectic Sturm theory.Ann. Fac. Sci. Toulouse, VI. Sér., Math. 2 6 (1993), 73-96. Zbl 0780.34004, MR 1230706, 10.5802/afst.758 |
| Reference:
|
[19] Ovsienko, V., Tabachnikov, S.: Projective Differential Geometry Old and New. From the Schwarzian Derivative to the Cohomology of Diffeomorphism Groups.Cambridge Tracts in Mathematics 165 Cambridge University Press, Cambridge (2005). Zbl 1073.53001, MR 2177471 |
| Reference:
|
[20] Radul, A. O.: Superstring Schwarz derivative and Bott cocycles.Integrable and Superintegrable Systems 336-351 World. Sci. Publ. Teaneck B. Kupershmidt (1990). MR 1091271 |
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