THE PRIME IDEALS AND SIMPLE MODULES OF THE UNIVERSAL ENVELOPING ALGEBRA U(b⋉V2)

BAVULA, VOLODYMYR V. ; LU, TAO

Glasgow mathematical journal, Tome 62 (2020), pp. S77-S98

Résumé

Let b be the Borel subalgebra of the Lie algebra sl2 and V2 be the simple two-dimensional sl2-module. For the universal enveloping algebra $\[{\cal A}: = U(\gb \ltimes {V_2})\]$ of the semi-direct product b⋉V2 of Lie algebras, the prime, primitive and maximal spectra are classified. Please approve edit to the sentence “The sets of completely prime...”.The sets of completely prime ideals of $\[{\cal A}\]$ are described. The simple unfaithful $\[{\cal A}\]$-modules are classified and an explicit description of all prime factor algebras of $\[{\cal A}\]$ is given. The following classes of simple U(b⋉V2)-modules are classified: the Whittaker modules, the K[X]-torsion modules and the K[E]-torsion modules.

Détail

DOI : 10.1017/S0017089519000302

Bibliographie

[1] Bavula, V. V., The simple D[X, Y; σ, a]-modules, Ukrainian Math. J. 44 (1992), 1628–1644. Google Scholar | DOI

[2] Bavula, V. V., Generalized Weyl algebras and their representations, St. Petersburg Math. J. 4(1) (1993), 71–92. Google Scholar

[3] Bavula, V. V., Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras, Bull. Sci. Math. 120(3) (1996), 293–335. Google Scholar

[4] Bavula, V. V., Global dimension of generalized Weyl algebras, in Representation theory of algebras (Cocoyoc, 1994), CMS Conf. Proc., 18, American Mathematics Society, Providence, RI (1996) 81–107. Google Scholar

[5] Bavula, V. V., Classification of the simple modules of the quantum Weyl algebra and the quantum plane, in Quantum groups and quantum spaces (Warsan, 1995), vol. 40 (Banach Center Publ., Polish Acad. Sci., Warsaw, 1997), 193–201. Google Scholar

[6] Bavula, V. V. and Van Oystaeyen, F., The simple modules of certain generalized crossed products, J. Algebra 194 (1997), 521–566. Google Scholar | DOI

[7] Bavula, V. V., The simple modules of Ore extensions with coefficients from a Dedekind ring, Comm. Algebra 27(6) (1999), 2665–2699. Google Scholar | DOI

[8] Bavula, V. and Van Oystaeyen, F., Simple modules of the Witten–Woronowicz algebra, J. Algebra 271 (2004), 827–845. Google Scholar | DOI

[9] Bavula, V. V. and Lu, T., The prime spectrum and simple modules over the quantum spatial ageing algebra, Algebr. Represent. Theory 19 (2016), 1109–1133. Google Scholar | DOI

[10] Benkart, G., Lopes, S. A. and Ondrus, M., A parametric family of subalgebras of the Weyl algebra II. Irreducible modules, in Algebraic and combinatorial approaches to representation theory (Chari, V., Greenstein, J., Misra, K. C., Raghavan, K. N. and Viswanath, S., Editors), Contemp. Math., vol. 602 (American Mathematics Society, Providence, RI, 2013), 73–98. Google Scholar

[11] Block, R. E., The irreducible representations of the Lie algebra (2) and of the Weyl algebra, Adv. Math. 39 (1981), 69–110. Google Scholar | DOI

[12] Henkel, M. and Stoimenov, S., On non-local representations of the ageing algebra, Nuclear Phys. B 847(3) (2011), 612–627. Google Scholar

[13] Lü, R., Mazorchuk, V. and Zhao, K., Classification of simple weight modules over the 1-spatial ageing algebra, Algebr. Represent. Theory 18(2) (2015), 381–395. Google Scholar | DOI

[14] Mcconnell, J. C. and Robson, J. C., Noncommutative noetherian rings, Graduate Studies in Mathematics, vol. 30 (American Mathematical Society, Providence, RI, 2001). Google Scholar

[15] Sigurdsson, G., Differential operator rings whose prime factors have bounded Goldie dimension, Arch. Math. (Basel) 42(4) (1984), 348–353. Google Scholar

[16] Stoimenov, S. and Henkel, M., Non-local space-time transformations generated from the ageing algebra, in Lie theory and its applications in physics (Dobrev, V., Editor) Springer Proceedings in Mathematics & Statistics, vol. 36 (Springer, Tokyo, 2013), 369–379. Google Scholar | DOI

[17] Stoimenov, S. and Henkel, M., Non-local representations of the ageing algebra in higher dimensions, J. Phys. A 46(24) (2013), 245004, 18pp. Google Scholar | DOI