| Title:
|
Formal deformations and principal series representations of ${\rm SL}(2,{\mathbb R})$ and ${\rm SL}(2,{\mathbb C})$ (English) |
| Author:
|
Cahen, Benjamin |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
70 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
935-951 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this note, we study formal deformations of derived representations of the principal series representations of ${\rm SL}(2,{\mathbb R})$. In particular, we recover all the representations of the derived principal series by deforming one of them. Similar results are also obtained for ${\rm SL}(2,{\mathbb C})$. (English) |
| Keyword:
|
deformation of representation |
| Keyword:
|
Lie algebra |
| Keyword:
|
Chevalley-Eilenberg cohomology |
| Keyword:
|
Moyal star product |
| Keyword:
|
Weyl correspondence |
| Keyword:
|
minimal realization |
| MSC:
|
17B10 |
| MSC:
|
17B20 |
| MSC:
|
17B56 |
| MSC:
|
22E46 |
| MSC:
|
53D55 |
| idZBL:
|
07285971 |
| idMR:
|
MR4181788 |
| DOI:
|
10.21136/CMJ.2020.0053-19 |
| . |
| Date available:
|
2020-11-18T09:42:13Z |
| Last updated:
|
2023-01-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148403 |
| . |
| Reference:
|
[1] Arnal, D., Benamor, H., Cahen, B.: Minimal realizations of classical simple Lie algebras through deformations.Ann. Fac. Sci. Toulouse, VI. Sér., Math. 7 (1998), 169-184. Zbl 0923.17010, MR 1656166, 10.5802/afst.894 |
| Reference:
|
[2] Cahen, B.: Construction par déformation de réalisations minimales d'une algèbre de Lie simple de type $G_2$.C. R. Acad. Sci., Paris, Sér. I 323 (1996), 853-857 French corrigendum ibid. 355 485 2017. Zbl 0864.17010, MR 1414547 |
| Reference:
|
[3] Cahen, B.: Deformation program for principal series representations.Lett. Math. Phys. 36 (1996), 65-75. Zbl 0843.22020, MR 1371298, 10.1007/BF00403252 |
| Reference:
|
[4] Cahen, B.: Déformations formelles de certaines représentations de l'algèbre de Lie d'un groupe de Poincaré généralisé.Ann. Math. Blaise Pascal 8 (2001), 17-37 French. Zbl 0987.22008, MR 1863645, 10.5802/ambp.133 |
| Reference:
|
[5] Cahen, B.: Weyl quantization for semidirect products.Differ. Geom. Appl. 25 (2007), 177-190. Zbl 1117.81087, MR 2311733, 10.1016/j.difgeo.2006.08.005 |
| Reference:
|
[6] Cahen, B.: Berezin quantization and holomorphic representations.Rend. Semin. Mat. Univ. Padova 129 (2013), 277-297. Zbl 1272.22007, MR 3090642, 10.4171/RSMUP/129-16 |
| Reference:
|
[7] Cahen, B.: A construction by deformation of unitary irreducible representations of $SU(1,n)$ and $SU(n+1)$.J. Algebra Appl. 18 (2019), Article ID 1950125, 15 pages. Zbl 07096440, MR 3977786, 10.1142/S0219498819501251 |
| Reference:
|
[8] Combescure, M., Robert, D.: Coherent States and Applications in Mathematical Physics.Theoretical and Mathematical Physics. Springer, Berlin (2012). Zbl 1243.81004, MR 2952171, 10.1007/978-94-007-0196-0 |
| Reference:
|
[9] Fialowski, A.: Deformations in mathematics and physics.Int. J. Theor. Phys. 47 (2008), 333-337. Zbl 1140.81416, MR 2396768, 10.1007/s10773-007-9454-7 |
| Reference:
|
[10] Fialowski, A., Montigny, M. de: Deformations and contractions of Lie algebras.J. Phys. A, Math. Gen. 38 (2005), 6335-6349. Zbl 1073.22012, MR 2166626, 10.1088/0305-4470/38/28/006 |
| Reference:
|
[11] Fialowski, A., Penkava, M.: The moduli space of 4-dimensional nilpotent complex associative algebras.Linear Algebra Appl. 457 (2014), 408-427. Zbl 1294.14008, MR 3230454, 10.1016/j.laa.2014.05.014 |
| Reference:
|
[12] Folland, G. B.: Harmonic Analysis in Phase Space.Annals of Mathematics Studies 122. Princeton University Press, Princeton (1989). Zbl 0682.43001, MR 0983366, 10.1515/9781400882427 |
| Reference:
|
[13] Gerstenhaber, M.: On the deformation of rings and algebras.Ann. Math. 79 (1964), 59-103. Zbl 0123.03101, MR 0171807, 10.2307/1970484 |
| Reference:
|
[14] Guichardet, A.: Cohomologie des Groupes Topologiques et des Algèbres de Lie.Textes Mathématiques 2. Cedic, Paris (1980), French. Zbl 0464.22001, MR 0644979 |
| Reference:
|
[15] Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces.Graduate Studies in Mathematics 34. American Mathematical Society, Providence (2001). Zbl 0993.53002, MR 1834454, 10.1090/gsm/034 |
| Reference:
|
[16] Hermann, R.: Analytic continuation of group representations. IV.Commun. Math. Phys. 5 (1967), 131-156. Zbl 0144.46105, MR 0231948, 10.1007/BF01646842 |
| Reference:
|
[17] Hörmander, L.: The Analysis of Linear Partial Differential Operators. III.Grundlehren der Mathematischen Wissenschaften 274. Springer, Berlin (1985). Zbl 0601.35001, MR 0781536, 10.1007/978-3-540-49938-1 |
| Reference:
|
[18] Joseph, A.: Minimal realizations and spectrum generating algebras.Commun. Math. Phys. 36 (1974), 325-338. Zbl 0285.17007, MR 0342049, 10.1007/BF01646204 |
| Reference:
|
[19] Kirillov, A. A.: Lectures on the Orbit Method.Graduate Studies in Mathematics 64. American Mathematical Society, Providence (2004). Zbl 1229.22003, MR 2069175, 10.1090/gsm/064 |
| Reference:
|
[20] Knapp, A. W.: Representation Theory of Semisimple Groups: An Overview Based on Examples.Princeton Mathematical Series 36. Princeton University Press, Princeton (1986). Zbl 0604.22001, MR 0855239, 10.1515/9781400883974 |
| Reference:
|
[21] Lesimple, M., Pinczon, G.: Deformations of Lie group and Lie algebra representations.J. Math. Phys. 34 (1993), 4251-4272. Zbl 0798.22009, MR 1233270, 10.1063/1.529998 |
| Reference:
|
[22] Levy-Nahas, M.: Deformation and contraction of Lie algebras.J. Math. Phys. 8 (1967), 1211-1222. Zbl 0175.24803, MR 0224747, 10.1063/1.1705338 |
| Reference:
|
[23] Levy-Nahas, M., Seneor, R.: First order deformations of Lie algebras representations, $E(3)$ and Poincaré examples.Commun. Math. Phys. 9 (1968), 242-266. Zbl 0161.46003, MR 0235804, 10.1007/BF01645689 |
| Reference:
|
[24] A. Nijenhuis, R. W. Richardson, Jr.: Cohomology and deformations in graded Lie algebras.Bull. Am. Math. Soc. 72 (1966), 1-29. Zbl 0136.30502, MR 0195995, 10.1090/S0002-9904-1966-11401-5 |
| Reference:
|
[25] A. Nijenhuis, R. W. Richardson, Jr.: Deformations of homomorphisms of Lie groups and Lie algebras.Bull. Am. Math. Soc. 73 (1967), 175-179. Zbl 0153.04402, MR 0204575, 10.1090/S0002-9904-1967-11703-8 |
| Reference:
|
[26] A. Nijenhuis, R. W. Richardson, Jr.: Deformations of Lie algebras structures.J. Math. Mech. 17 (1967), 89-105. Zbl 0166.30202, MR 0214636, 10.1512/iumj.1968.17.17005 |
| Reference:
|
[27] Voros, A.: An algebra of pseudo differential operators and the asymptotics of quantum mechanics.J. Funct. Anal. 29 (1978), 104-132. Zbl 0386.47031, MR 0496088, 10.1016/0022-1236(78)90049-6 |
| Reference:
|
[28] Wallach, N. R.: Harmonic Analysis on Homogeneous Spaces.Pure and Applied Mathematics 19. Marcel Dekker, New York (1973). Zbl 0265.22022, MR 0498996 |
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