La dimension de Krull de U(s\(\ell (3))\). (The Krull dimension of U(s\(\ell (3)))\)
type: Item
instance of: scholarly article
title: La dimension de Krull de U(s\(\ell (3))\). (The Krull dimension of U(s\(\ell (3)))\)
zbMATH Open document ID: 0595.17006
DOI: 10.1016/0021-8693(86)90127-4
author: Thierry Levasseur
published in: Journal of Algebra
publication date: 1986-01-01T00:00:00Z
review text: It is proved that the Krull dimension of the enveloping algebra of \(s\ell (3,{\mathbb{C}})\), denoted U(s\(\ell (3))\), is equal to 5. For a general semisimple Lie algebra \({\mathfrak g}\), it is conjectured that the Krull dimension of U(\({\mathfrak g})\) equals the dimension of a Borel subalgebra of \({\mathfrak g}\). It is easy to establish this as a lower bound. The problem is to obtain the upper bound. Interesting ideas are used to solve this problem for \(s\ell (3)\). It is to be expected that equally interesting ideas will be required to settle the conjecture. Let \(G=SL(3,{\mathbb{C}})\), and let N be the unipotent subgroup of upper triangular matrices. The proof is in part based on earlier work of the author [Lect. Notes Math. 924, 173-183 (1982; Zbl 0478.17003)] showing that the Krull dimension of U(s\(\ell (3))\) is bounded above by the Krull dimension of \({\mathcal D}(G/N)\), the ring of differential operators on the homogeneous space G/N. However, \({\mathcal O}(G/N)\cong {\mathbb{C}}[X_ 1,...,X_ 6]/(X^ 2_ 1+...+X^ 2_ 6)\) is not a regular ring, so it is not easy to describe \({\mathcal D}(G/N)\). This difficulty is overcome by using a result of \textit{A. B. Goncharov} [Funkts. Anal. Prilozh. 15, No.3, 83-84 (1981; Zbl 0485.53032)] which shows that the Lie algebra so(8) acts as differential operators on the quadratic cone in \({\mathbb{C}}^ 6\) defined by Max Spec \({\mathcal O}(G/N)\). The kernel of the map \(\psi\) : U(so(8))\(\to {\mathcal D}(G/N)\) is the Joseph ideal, hence the Krull dimension of the image of \(\psi\) is already known to be 5 [the reviewer, Math. Proc. Camb. Philos. Soc. 93, 459-466 (1983; Zbl 0535.17008)]. Finally, the relationship between the image of \(\psi\) and \({\mathcal D}(G/N)\) is sufficiently ''close'' to allow one to conclude that these algebras have the same Krull dimension.
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 16P60
Mathematics Subject Classification ID: 17B20
zbMATH DE Number: 3957377
zbMATH Keywords: Krull dimension
zbMATH Keywords: enveloping algebra
zbMATH Keywords: semisimple Lie algebra
zbMATH Keywords: s\(\ell (3)\)
zbMATH Keywords: Joseph ideal
reviewed by: S. Paul Smith
MaRDI profile type: Publication
full work available at URL: 0021-8693(86)90127-4
OpenAlex ID: W1981699411
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description: scientific article; zbMATH DE number 3957377