Orbits and primitive ideals of solvable Lie algebras
type: Item
instance of: scholarly article
title: Orbits and primitive ideals of solvable Lie algebras
zbMATH Open document ID: 0788.17008
DOI: 10.1007/BF01459737
author: Niels Vigand Pedersen
published in: Mathematische Annalen
publication date: 1994-06-06T00:00:00Z
full work available at URL: https://eudml.org/doc/165173
review text: Let \({\mathfrak g}\) be a complex solvable Lie algebra, and let \(U({\mathfrak g})\) be the universal enveloping algebra of \({\mathfrak g}\). The Dixmier map \(I:{\mathfrak g}^*\to \text{Prim}(U({\mathfrak g}))\) associates with each element \(g\) in the dual \({\mathfrak g}^*\) of \({\mathfrak g}\) a primitive ideal \(I(g)\) of \(U({\mathfrak g})\). This map is surjective, and the inverse image by \(I\) of \(I(g)\) is the orbit \(O\) through \(g\) of the algebraic adjoint \({\mathcal G}\) of \({\mathfrak g}\), i.e., the factorized Dixmier map \(\overline{I}: {\mathfrak g}^*/{\mathcal G} \to \text{Prim}(U({\mathfrak g}))\), where \({\mathfrak g}^*/{\mathcal G}\) is the space of \({\mathcal G}\)-orbits, is a bijection [\textit{J. Dixmier}, J. Math. Pures Appl., IX. Sér. 45, 1-66 (1966; Zbl 0136.306); \textit{M. Duflo}, Ann. Sci. Éc. Norm. Supér. 5, 71-120 (1972; Zbl 0241.22030); \textit{R. Rentschler}, Invent. Math. 23, 49- 71 (1974; Zbl 0299.17003)]. The Dixmier map is established using induction (in its Lie algebra version) of one dimensional representations of polarizations. The main purpose of this paper is to establish a direct link between orbits and primitive ideals of \({\mathfrak g}\). This is done by giving a detailed explicit description of the orbits of the algebraic adjoint of \({\mathfrak g}\), and then showing that an analogous description of the primitive ideals is possible. This shows that there is a much closer connection between orbits and primitive ideals than previously known. The author applies his results to give an explicit description of the inverse of the factorized Dixmier map. In his previous publication [ Bull. Soc. Math. Fr. 112, 423-467 (1984; Zbl 0589.22009)] the author showed how to construct a (finite) set of generators for the infinitesimal kernel of irreducible representations of a nilpotent Lie group (which is a primitive ideal of \(U({\mathfrak g})\); see also \textit{C. Godfrey}, Trans. Am. Math. Soc. 233, 295-307 (1977; Zbl 0…
reviewed by: Niels Vigand Pedersen
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 22E27
Mathematics Subject Classification ID: 17B30
zbMATH DE Number: 537319
zbMATH Keywords: complex solvable Lie algebra
zbMATH Keywords: universal enveloping algebra
zbMATH Keywords: Dixmier map
zbMATH Keywords: primitive ideal
zbMATH Keywords: factorized Dixmier map
zbMATH Keywords: orbits
zbMATH Keywords: jump index stratification
zbMATH Keywords: filtrations
MaRDI profile type: Publication
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OpenAlex ID: W2024845951
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description: scientific article; zbMATH DE number 537319