Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-10T12:08:47.487Z Has data issue: false hasContentIssue false
  • English
  • Français

The Universal Enveloping Algebra of the Schrödinger Algebra and its Prime Spectrum

Published online by Cambridge University Press:  20 November 2018

V. V. Bavula  and
T. Lu
V. V. Bavula
Affiliation:
Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield, S3 7RH, UK, e-mail : v.bavula@sheffield.ac.uk
T. Lu
Affiliation:
School of Mathematical Sciences, Huaqiao University, Quanzhou, Fujian 362021, China, e-mail : lutao@hqu.edu.cn
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The prime, completely prime, maximal, and primitive spectra are classified for the universal enveloping algebra of the Schrödinger algebra. The explicit generators are given for all of these ideals. A counterexample is constructed to the conjecture of Cheng and Zhang about nonexistence of simple singular Whittaker modules for the Schrödinger algebra (and all such modules are classified). It is proved that the conjecture holds ‘generically’.

Keywords

17B1016D2516D6016D7016P50prime idealweight modulesimple modulecentralizerWhittaker module
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 61 , Issue 4 , 01 December 2018 , pp. 688 - 703
Copyright
Copyright © Canadian Mathematical Society 2018

References

[1] Aizawa, N. and Dobrev, V., Interwining operator realization of non-relativistic holography. Nuclear Phys. B 828 (2010), 581–593. http://dx.doi.Org/10.1016/j.nuclphysb.2OO9.10.019Google Scholar
[2] Bavula, V. V., Description of two-sided ideals in a class of noncommutative rings. I. Ukrainian Math. J. 45 (1993) no. 2, 223–234. http://dx.doi.org/10.1007/BF01060977Google Scholar
[3] Bavula, V. V., Description of two-sided ideals in a class of noncommutative rings. II. Ukrainian Math. J. 45 (1993) no. 3, 329–334. http://dx.doi.org/10.1007/BF01061007Google Scholar
[4] Bavula, V. V., The simple modules of Ore extensions with coefficients from a Dedekind ring. Comm. Algebra 27 (1999) no. 6, 2665–2699. http://dx.doi.org/10.1080/00927879908826587Google Scholar
[5] Bavula, V. V. and Lu, T., The universal enveloping algebra U(sl2 ⋉ V2), its prime spectrum and a classification of its simple weight modules. J. Lie Theory 28 (2018) no. 2, 525–560.Google Scholar
[6] Bavula, V. V. and Lu, T., The prime spectrum and simple modules over the quantum spatial ageing algebra. Algebra Represent. Theory 19 (2016) no. 5, 1109–1133. http://dx.doi.org/10.1007/s10468-016-9613-8Google Scholar
[7] Bavula, V. V. and Lu, T., Classification of simple weight modules over the Schrodinger algebra. Canad. Math. Bull. 61 (2018), 16–39. http://dx.doi.org/10.4153/CMB-2017-017-7Google Scholar
[8] Block, R. E., The irreducible representations of the Lie algebra sl(2) and of the Weyl algebra. Adv. in Math. 39 (1981) no. 1, 69–110. http://dx.doi.Org/10.1016/0001-8708(81)90058-XGoogle Scholar
[9] Cai, Y., Cheng, Y. and Shen, R., Quasi-Whittaker modules for the Schrodinger algebra. Linear Algebra Appl. 463 (2014), 16–32. http://dx.doi.Org/10.1016/j.laa.2014.09.001Google Scholar
[10] Dixmier, J., Enveloping algebras. Graduate Studies in Mathematics, 11, American Mathematical Society, Providence, RI, 1996. http://dx.doi.Org/10.1090/gsm/011Google Scholar
[11] Dubsky, B., Classification of simple weight modules with finite-dimensional weight spaces over the Schrodinger algebra. Linera Algebra Appl. 443 (2014), 204–214. http://dx.doi.Org/10.1016/j.laa.2O13.11.016Google Scholar
[12] Dubsky, B., Lii, R., Mazorchuk, V., and Zhao, K., Category O for the Schrodinger algebra. Linear Algebra Appl. 460 (2014), 17–50. http://dx.doi.Org/10.1016/j.laa.2O14.07.030Google Scholar
[13] Dobrev, V., Doebner, H. D., and Mrugalla, C., Lowest weight representations of the Schrodinger algebra and generalized heat/Schrb'dinger equations. Rep. Math. Phys. 39 (1997), 201–218. http://dx.doi.Org/10.1016/S0034-4877(97)88001-9Google Scholar
[14] McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings. Graduate Studies in Mathematics, 30, American Mathematical Society, Providence, RI. 2001.Google Scholar
[15] Perroud, M., Projective representations of the Schrodinger group. Helv. Phys. Acta 50 (1977) no. 2, 233–252.Google Scholar
[16] Wu, Y., Finite dimensional indecomposable modules for Schrodinger algebra. J. Math. Phys. 54 (2013), 073503. http://dx.doi.Org/10.1063/1.4812558Google Scholar
[17] Wu, Y. and Zhu, L., Simple weight modules for Schrodinger algebra. Linear Algebra Appl. 438 (2013), 559–563. http://dx.doi.Org/10.1016/j.laa.2O12.07.029Google Scholar
[18] Zhang, X. and Cheng, Y., Simple Schrodinger 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.2014.09.029Google Scholar