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On supports and associated primes of modules over the enveloping algebras of nilpotent Lie algebras


Author: Boris Sirola
Journal: Trans. Amer. Math. Soc. 353 (2001), 2131-2170
MSC (2000): Primary 17B35, 16P50
DOI: https://doi.org/10.1090/S0002-9947-01-02741-6
Published electronically: January 3, 2001
MathSciNet review: 1814065
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Abstract:

Let $\mathfrak n$ be a nilpotent Lie algebra, over a field of characteristic zero, and $\mathcal U$ its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of $\mathcal U$ related to finitely generated $\mathcal U$-modules $V$, and in particular the set $\operatorname{Ass}V$ of associated primes for such $V$ (note that now $\operatorname{Ass}V$ is equal to the set $\operatorname{Annspec}V$ of annihilator primes for $V$); (2) the problem of nontriviality for the modules $V/\mathcal PV$ when $\mathcal P$ is a (maximal) prime of $\mathcal U$, and in particular when $\mathcal P$ is the augmentation ideal $\mathcal U\mathfrak n$ of $\mathcal U$. We define the support of $V$, as a natural generalization of the same notion from commutative theory, and show that it is the object of primary interest when dealing with (2). We also introduce and study the reduced localization and the reduced support, which enables to better understand the set $\operatorname{Ass}V$. We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when $\mathfrak N$, $\mathfrak N$ as in the theorem, are abelian. We also present some of its interesting consequences.

Theorem. Let $\mathfrak Q$ be a finite-dimensional Lie algebra over a field of characteristic zero, and $\mathfrak N$ an ideal of $\mathfrak Q$; denote by $U(\mathfrak N)$ the universal enveloping algebra of $\mathfrak N$. Let $V$ be a $\mathfrak Q$-module which is finitely generated as an $\mathfrak N$-module. Then every annihilator prime of $V$, when $V$ is regarded as a $U(\mathfrak N)$-module, is $\mathfrak Q$-stable for the adjoint action of $\mathfrak Q$ on $U(\mathfrak N)$.


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  • [BB] A. Beilinson and J. Bernstein, A generalization of Casselman's submodule theorem, Representation Theory of Reductive Groups (Peter C. Trombi, ed.), Progress in Mathematics, vol. 40, Birkhäuser, Boston, 1983, pp. 35-52. MR 85e:22024
  • [Bh] W. Borho, A survey on enveloping algebras of semisimple Lie algebras, Lie Algebras and Related Topics, CMS Conf. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 1986, pp. 19-50. MR 87g:17013
  • [BGR] W. Borho, P. Gabriel and R. Rentschler, Primideale in Einhüllenden auflösbarer Lie-Algebren, Lecture Notes in Math. 357, Springer-Verlag, Berlin, 1973. MR 51:12965
  • [B1] N. Bourbaki, Commutative Algebra, Hermann, Paris, and Addison Wesley, Reading, Massachusetts, 1972. MR 50:12997
  • [B2] -, Algèbre, Chapitres I,II,III, Chapitres VII,VIII, Hermann, Paris, 1970. MR 43:2
  • [B3] -, Groupes et Algèbres de Lie, Chapitres IV,V,VI, Hermann, Paris, 1968. MR 39:1590
  • [C] W. Casselman, Jacquet modules for real reductive groups, Proc. Int. Cong. Math., Helsinki (1978), pp. 557-563. MR 83h:22025
  • [CM] W. Casselman and D. Milicic, Asymptotic behavior of matrix coefficients of admissible representations, Duke Math. J. $\mathbf{49}$ (1982), 869-930. MR 85a:22024
  • [CO] W. Casselman and M. S. Osborne, The restriction of admissible representations to $\mathfrak n$, Math. Ann. $\mathbf{233}$ (1978), 193-198 MR 58:1033.
  • [D] J. Dixmier, Enveloping Algebras, North-Holland Mathematical Library, Vol. 14, North-Holland, Amsterdam, 1977. MR 58:16803b
  • [GW] K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, ``London Math. Soc. Stud. Texts", Vol. 16, Cambridge Univ. Press, Cambridge, 1989. MR 91c:16001
  • [HT] R. Howe and E. C. Tan, Non-Abelian Harmonic Analysis, Applications of ${SL}(2,\mathbb R)$, Springer-Verlag, New York, 1992. MR 93f:22009
  • [J] A. V. Jategaonkar, Localization in Noetherian Rings, ``London Math. Soc. Lect. Note Ser.'', Vol. 98, Cambridge Univ. Press, Cambridge, 1986. MR 88c:16005
  • [K] A. W. Knapp, Lie Groups Beyond an Introduction, Progress in Mathematics, vol. 140, Birkhäuser, Boston, 1996. MR 98b:22002
  • [McC] J. C. McConnell, The intersection theorem for a class of non-commutative rings, Proc. London Math. Soc. (3) $\mathbf{17}$ (1967), 487-498. MR 35:1624
  • [MR] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, J. Wiley, New York, 1987. MR 89j:16023
  • [NG] Y. Nouazé and P. Gabriel, Idéaux premiers de l'algèbre enveloppante d'une algèbre de Lie nilpotente, J. Algebra $\mathbf{6}$ (1967), 77-99. MR 34:5889
  • [S] P. F. Smith, Localization and the AR property, Proc. London Math. Soc. (3) $\mathbf{22}$ (1971), 39-68. MR 45:3453
  • [St] J. T. Stafford, On the regular elements of Noetherian rings, Ring Theory, Proceedings of the 1978 Antwerp Conference (F. Van Oystaeyen, ed.), Lecture Notes in Pure and Appl. Math., vol. 51, Marcel Dekker, New York, 1979, pp. 257-277. MR 81k:16016
  • [SW] J. T. Stafford and N. R. Wallach, The restriction of admissible modules to parabolic subalgebras, Trans. Amer. Math. Soc. $\mathbf{272}$ (1982), 333-350. MR 83h:17007
  • [S1] B. Sirola, On associated primes and weakly associated primes, Interactions Between Ring Theory and Representations of Algebras, Proceedings of the 1998 Murcia Conference (F. Van Oystaeyen and M. Saorin, eds.), M. Dekker, New York (2000), pp. 383-391. CMP 2000:13
  • [S2] -, Annihilator primes and foundation primes, preprint.
  • [S3] -, Going up for the enveloping algebras of Lie algebras, submitted.
  • [S4] -, On noncommutative Noetherian schemes, preprint.
  • [V1] D. A. Vogan, Jr., The orbit method and primitive ideals for semisimple Lie algebras, Lie Algebras and Related Topics, CMS Conf. Proc., vol. 5, Amer. Math. Soc., Providence, RI, 1986, pp. 281-316. MR 87k:17015
  • [V2] -, Associated varieties and unipotent representations, Harmonic Analysis on Reductive Groups (W. Barker and P. Sally, eds.), Birkhäuser, Boston, 1991, pp. 315-388. MR 93k:22012

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Additional Information

Boris Sirola
Affiliation: Department of Mathematics, University of Zagreb, PP 335, 10002 Zagreb, Croatia
Email: sirola@math.hr

DOI: https://doi.org/10.1090/S0002-9947-01-02741-6
Keywords: Nilpotent Lie algebra, semisimple Lie algebra, parabolic subalgebra, universal enveloping algebra, prime ideal, augmentation ideal, associated prime, annihilator prime, foundation prime, localization, support, reduced localization, reduced support, (co)adjoint action, stable prime, coadjoint orbit
Received by editor(s): February 5, 1999
Received by editor(s) in revised form: June 3, 1999, and January 31, 2000
Published electronically: January 3, 2001
Additional Notes: The author was supported by Fulbright Grant No. 22676, and in part by the Ministry of Science and Technology, Republic of Croatia
Article copyright: © Copyright 2001 American Mathematical Society

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