Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Ideals of coadjoint orbits of nilpotent Lie algebras


Author: Colin Godfrey
Journal: Trans. Amer. Math. Soc. 233 (1977), 295-307
MSC: Primary 17B30
MathSciNet review: 0447359
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For f a linear functional on a nilpotent Lie algebra g over a field of characteristic 0, let $ J(f)$ be the ideal of all polynomials in $ S(g)$ vanishing on the coadjoint orbit through f in $ {g^\ast}$, and let $ I(f)$ be the primitive ideal of Dixmier in the universal enveloping algebra $ U(g)$, corresponding to the orbit. An inductive method is given for computing generators $ {P_1}, \ldots ,{P_r}$ of $ J(f)$ such that $ \varphi {P_1}, \ldots ,\varphi {P_r}$ generate $ I(f),\varphi $ being the symmetrization map from $ S(g)$ to $ U(g)$. Upper bounds are given for the number of variables in the polynomials $ {P_i}$ and a counterexample is produced for upper bounds proposed by Kirillov.


References [Enhancements On Off] (What's this?)

  • [1] Nicole Conze and Jacques Dixmier, Idéaux primitifs dans l’algèbre enveloppante d’une algèbre de Lie semi-simple, Bull. Sci. Math. (2) 96 (1972), 339–351 (French). MR 0321991
  • [2] Jacques Dixmier, Algèbres enveloppantes, Gauthier-Villars Éditeur, Paris-Brussels-Montreal, Que., 1974 (French). Cahiers Scientifiques, Fasc. XXXVII. MR 0498737
  • [3] Jacques Dixmier, Sur les représentations unitaries des groupes de Lie nilpotents. III, Canad. J. Math. 10 (1958), 321–348. MR 0095427
  • [4] J. Dixmier, Représentations irréductibles des algèbres de Lie nilpotentes, An. Acad. Brasil. Ci. 35 (1963), 491–519 (French). MR 0182682
  • [5] C. Godfrey, Thesis, Harvard Univ., 1974
  • [6] A. A. Kirillov, Unitary representations of nilpotent Lie groups, Uspehi Mat. Nauk 17 (1962), no. 4 (106), 57–110 (Russian). MR 0142001
  • [7] Bertram Kostant, Lie group representations on polynomial rings, Amer. J. Math. 85 (1963), 327–404. MR 0158024
  • [8] Y. Nouazé and P. Gabriel, Idéaux premiers de l’algèbre enveloppante d’une algèbre de Lie nilpotente, J. Algebra 6 (1967), 77–99 (French). MR 0206064
  • [9] M. Vergne, Thesis, Université de Paris, 1966.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 17B30

Retrieve articles in all journals with MSC: 17B30


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1977-0447359-9
Keywords: Coadjoint orbits, nilpotent Lie algebra, primitive ideal, symmetrization map, universal enveloping algebra
Article copyright: © Copyright 1977 American Mathematical Society