Geodesic
Classification of Simple Weight Modules over the Schrödinger Algebra
Canadian mathematical bulletin, Tome 61 (2018) no. 1, pp. 16-39

Voir la notice de l'article provenant de la source Cambridge University Press

A classification of simple weight modules over the Schrödinger algebra is given. The Krull and the global dimensions are found for the centralizer ${{C}_{S}}(H)$ (and some of its prime factor algebras) of the Cartan element $H$ in the universal enveloping algebra $S$ of the Schrödinger (Lie) algebra. The simple ${{C}_{S}}(H)$ -modules are classified. The Krull and the global dimensions are found for some (prime) factor algebras of the algebra $S$ (over the centre). It is proved that some (prime) factor algebras of $S$ and ${{C}_{S}}(H)$ are tensor homological $/$ Krull minimal.
DOI : 10.4153/CMB-2017-017-7
Mots-clés : 17B10, 17B20, 17B35, 16E10, 16P90, 16P40, 16P50, weight module, simple module, centralizer, Krull dimension, global dimension, tensor homological minimal algebra, tensor Krull minimal algebra 
Bavula, V. V.; Lu, T. Classification of Simple Weight Modules over the Schrödinger Algebra. Canadian mathematical bulletin, Tome 61 (2018) no. 1, pp. 16-39. doi: 10.4153/CMB-2017-017-7
@article{10_4153_CMB_2017_017_7,
     author = {Bavula, V. V. and Lu, T.},
     title = {Classification of {Simple} {Weight} {Modules} over the {Schr\"odinger} {Algebra}},
     journal = {Canadian mathematical bulletin},
     pages = {16--39},
     year = {2018},
     volume = {61},
     number = {1},
     doi = {10.4153/CMB-2017-017-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-017-7/}
}
TY  - JOUR
AU  - Bavula, V. V.
AU  - Lu, T.
TI  - Classification of Simple Weight Modules over the Schrödinger Algebra
JO  - Canadian mathematical bulletin
PY  - 2018
SP  - 16
EP  - 39
VL  - 61
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-017-7/
DO  - 10.4153/CMB-2017-017-7
ID  - 10_4153_CMB_2017_017_7
ER  - 
%0 Journal Article
%A Bavula, V. V.
%A Lu, T.
%T Classification of Simple Weight Modules over the Schrödinger Algebra
%J Canadian mathematical bulletin
%D 2018
%P 16-39
%V 61
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2017-017-7/
%R 10.4153/CMB-2017-017-7
%F 10_4153_CMB_2017_017_7

[1] [1] Auslander, M., On the dimension ofmodules and algebras. III. Global dimension. Nagoya Math. J. 9 (1955), 66–77. http://dx.doi.org/10.1017/S0027763000023291. Google Scholar

[2] [2] Bavula, V. V., Finite-dimensionality of Extn and Tor of simple modules over a class of algebras. Funct. Anal. Appl. 25(1991) no. 3, 229–230. http://dx.doi.org/!0.1007/BF01085496. Google Scholar

[3] [3] Bavula, V. V., Simple D[X, Y; a, a]-modules. Ukrainian Math. J. 44 (1992), no. 12, 1500–1511. http://dx.doi.Org/10.1007/BF01061 275. Google Scholar

[4] [4] Bavula, V. V., Generalized Weyl algebras, kernet and tensor-simple algebras, their simple modules. In: Representations of algebras (Ottawa, 1992), CMS Conf. Proc, 14, American Mathematical Society, Providence, RI, 1993, pp. 83-107. Google Scholar

[5] [5] Bavula, V. V., Description oftwo-sided Ideals in a class of noncommutative rings. I. Ukrainian Math. J. 45 (1993), no. 2, 223–234. http://dx.doi.org/10.1007/BF01060977. Google Scholar

[6] [6] Bavula, V. V., Generalized Weyl algebras and their representations. St. Petersburg Math. J. 4 (1993), no. 1, 71–92. Google Scholar

[7] [7] Bavula, V. V., Global dimension of generalized Weyl algebras. In: Representation of Algebras (Cocoyoc, 1994), CMS Conf. Proc, 18, American Mathematical Society, Providence, RI, 1996, pp. 81-107. Google Scholar

[8] [8] Bavula, V. V., Tensor homological minimal algebras, global dimension ofthe tensor product of algebras and of generalized Weyl algebras. Bull. Sei. Math. 120 (1996), no. 3, 293–335. Google Scholar

[9] [9] Bavula, V. V. and Lenagan, T., Krull dimension of generalized Weyl algebras with noncommutative coefficients. J. Algebra 235 (2001), 315–358. http://dx.doi.Org/10.1006/jabr.2000.8466. Google Scholar

[10] [10] Bavula, V. V. and Lenagan, T., Generalized Weyl algebras are tensor Krull minimal. J. Algebra 239 (2001), 93–111. http://dx.doi.Org/10.1006/jabr.2000.8641. Google Scholar

[11] [11] Bavula, V. V. and Lu, T., The prime spectrum and simple modules over the quantum spatial ageing algebra. Algebr. Represent. Theory 19 (2016), no. 5, 1109–1133. http://dx.doi.org/10.1007/s10468-016-9613-8. Google Scholar

[12] [12] Bavula, V. V. and Lu, T., Prime ideals ofthe enveloping algebra ofthe Euclidean algebra and a classification ofits simple weight modules. J. Math. Phys. 58(2017), no. 1, 011701, 33. http://dx.doi.Org/10.1063/1.4973378. Google Scholar

[13] [13] Bavula, V. V. and Lu, T., Torsion simple modules over the quantum spatial ageing algebra. Commun. Algebra (published online 26 Oct. 2016). http://dx.doi.org/10.1080/00927872.2016.1240177. Google Scholar

[14] [14] Bavula, V. V. and Lu, T., The universal enveloping algebra U(sl 2 ⋉ V), its prime spectrum and a classification of its simple weight modules. J. Lie Theory, to appear. Google Scholar

[15] [15] Bavula, V. V. and Lu, T., The universal enveloping algebra ofthe Schrödinger algebra and its prime spectrum. Bull. Lond. Math. Soc, to appear. Google Scholar

[16] [16] Bavula, V. V. and Van Oystaeyen, F., Krull dimension of generalized Weyl algebras and iterated skew polynomial rings: commutative coefficients. J. Algebra 208 (1998), 1–34. http://dx.doi.org/10.1006/jabr.1998.7482. Google Scholar

[17] [17] Block, R. E., The irreducible representations ofthe Lie algebra s 1(2) and ofthe Weyl algebra. Adv. Math. 39 (1981), 69–110. http://dx.doi.Org/10.1016/0001-8708(81)90058-X Google Scholar

[18] [18] Dubsky, B., Classification of simple weight modules with finite-dimensional weight Spaces over the Schrödinger algebra. Linear Algebra Appl. 443 (2014), 204–214. http://dx.doi.Org/10.1016/j.laa.2O13.11.016. Google Scholar

[19] [19] Dubsky, B., Lü, R., Mazorchuk, V., and Zhao, K., Category O for the Schrödinger algebra. Linear Algebra Appl. 460 (2014), 17–50. http://dx.doi.Org/10.1016/j.laa.2O14.07.030. Google Scholar

[20] [20] Dobrev, V., Doebner, H. D., and Mrugalla, C., Lowest weight representations ofthe Schrödinger algebra and generalized heat/Schrödinger equations. Reports on Mathematical Physics 39 (1997), 201–218. http://dx.doi.Org/10.1016/S0034-4877(97)88001-9. Google Scholar

[21] [21] Hodges, T. J., Noncommutative deformation of type-A Kleinian singularities. J. Algebra 161(1993) no. 2, 271–290. http://dx.doi.org/10.1006/jabr.1993.1219. Google Scholar

[22] [22] Lü, R., Mazorchuk, V., and Zhao, K., Classification of simple weight modules over the 1-spatial ageing algebra. Algebr. Represent. Theory 18 (2015), no. 2, 381–395. http://dx.doi.Org/10.1007/s10468-014-9499-2. Google Scholar

[23] [23] McConnell, J. C. and Robson, J. C., Noncommutative noetherian rings. Graduate Studies in Mathematics, 30, American Mathematical Society, Providence, RI, 2001. http://dx.doi.Org/10.1090/gsm/030. Google Scholar

[24] [24] Perroud, M., Projective representations of the Schrödinger group. Helv. Phys. Acta 50 (1977), 233–252. Google Scholar

[25] [25] Reinhart, G. S., Note on the global dimension ofa certain ring. Proc. Amer. Math. Soc. 13 (1962), 341–346. http://dx.doi.org/10.1090/S0002-9939-1962-0137747-7. Google Scholar

[26] [26] Rentschier, R. and Gabriel, P., Sur la dimension des anneaux et ensembles ordonnes. C. R. Acad. Sei. Paris Ser. A-B, 265(1967), A712-A715. Google Scholar

[27] [27] Smith, S. P., Krull dimension ofthe enveloping algebra ofsl(2). J. Algebra 71 (1981), 89–94. http://dx.doi.Org/10.1016/0021-8693(81)90114-9. Google Scholar

[28] [28] Stafford, J. T., Homological properties ofthe enveloping algebra U(sl 2). Math. Proc. Camb. Phil. Soc. 91 (1982), no. 1, 29–37. http://dx.doi.org/10.1017/S0305004100059089 Google Scholar

[29] [29] Wu, Y. and Zhu, L., Simple weight modules for Schrödinger algebra. Linear Algebra Appl. 438 (2013), 559–563. http://dx.doi.Org/10.1016/j.laa.2O12.07.029. Google Scholar

[30] [30] Zhang, X. and Cheng, Y., Simple Schrödinger modules which are locally finite over the positive part. J. Pure Appl. Algebra 219 (2015), 2799–2815. http://dx.doi.Org/10.1016/j.jpaa.2O14.09.029 Google Scholar

Cité par Sources :