Actions of algebraic groups on the spectrum of rational ideals. II
type: Item
instance of: scholarly article
title: Actions of algebraic groups on the spectrum of rational ideals. II
zbMATH Open document ID: 0914.16013
author: Nikolaus Vonessen
published in: Journal of Algebra
publication date: 1999-06-14T00:00:00Z
review text: [For part I cf. ibid. 182, No. 2, 383-400 (1996; Zbl 0867.16020).] The author studies rational actions of a linear algebraic group \(G\) on an algebra \(V\), and the induced actions on \(\text{Rat}(V)\), the spectrum of rational ideals of \(V\) (a subset of \(\text{Spec}(V)\) which often includes all primitive ideals). This work extends results of Moeglin and Rentschler to prime characteristic, often also relaxing their finiteness assumptions on \(V\). In particular, some properties of the rational ideal \(J\) are related with its orb, the ideal \((J:G)=\bigcap_{\gamma\in G}\gamma(J)\). The rational ideals of \(V\) containing the orb of \(J\) are precisely those in the Zariski-closure \(X\) of the orbit of \(J\) in \(\text{Rat}(V)\). The \(G\)-stratum of \(J\) consists of all rational ideals in \(X\) whose orbit is dense in \(X\) (i.e. whose orb is equal to the orb of \(J\)). The author shows that the \(G\)-spectrum of a rational ideal consists of exactly one \(G\)-orbit, and that rational ideals are maximal in their strata in a strong sense. These results are useful for studying prime and primitive spectra of certain algebras, cf. recent work by \textit{K. R. Goodearl} and \textit{E. S. Letzter} [Trans. Am. Math. Soc. (to appear)]. It is shown further that the orbit of \(J\) is open in its closure in \(\text{Rat}(V)\), provided that \(J\) is locally closed. Among other results, the author proves that the semiprime ideal \((J:G)\) is Goldie, and relates the uniform and Gelfand-Kirillov dimensions of \(V/J\) and \(V/(J,G)\).
reviewed by: Fang Li
Mathematics Subject Classification ID: 16W20
Mathematics Subject Classification ID: 14L30
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 16D25
zbMATH DE Number: 1221662
zbMATH Keywords: Goldie condition
zbMATH Keywords: Gelfand-Kirillov dimensions
zbMATH Keywords: rational actions
zbMATH Keywords: linear algebraic groups
zbMATH Keywords: induced actions
zbMATH Keywords: spectrum of rational ideals
zbMATH Keywords: primitive ideals
zbMATH Keywords: orbits
zbMATH Keywords: semiprime ideals
MaRDI profile type: Publication
full work available at URL: jabr.1998.7422
OpenAlex ID: W2170546240
cites work: Some remarks on algebraic groups
cites work: Q5563439
cites work: Goldie Dimension of Prime Factors of Polynomial and Skew Polynomial Rings
cites work: Q3999894
cites work: Primideale in Einhüllenden auflösbarer Lie-Algebren (Beschreibung durch Bahnenräume)
cites work: Über die Gelfand-Kirillov-Dimension. (On the Gelfand-Kirillov-dimension)
cites work: Q4011048
cites work: Q4692705
cites work: Q3997571
cites work: Q4888783
cites work: Groups Acting on Affine Algebras
cites work: Prime and principal closed ideals in polynomial rings
cites work: A Note on Jacobson Rings and Polynomial Rings
cites work: The Dixmier-Moeglin equivalence in quantum coordinate rings and quantized Weyl algebras
cites work: Algebraic structure of multiparameter quantum groups
cites work: Q3136276
cites work: On the prime and primitive spectra of the algebra of functions on a quantum group
cites work: Q4317871
cites work: Q3677877
cites work: Goldie Conditions in Finite Normalizing Extensions
cites work: Q3786586
cites work: Orbites d'un groupe algébrique dans l'espace des idéaux rationnels d'une algèbre enveloppante
cites work: Sur la classification des idéaux primitifs des algèbres enveloppantes
cites work: Q1089086
cites work: Q4274976
cites work: Infinite Crossed Products and Group-Graded Rings
cites work: Q3795826
cites work: Liberal Extensions
cites work: Automorphisms of Function Fields
cites work: Q3910639
cites work: Actions of algebraic groups on the spectrum of rational ideals
cites work: Actions of linearly reductive groups on affine PI algebras
cites work: Q3884236
DOI: 10.1006/JABR.1998.7422
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description: scientific article; zbMATH DE number 1221662