La dimension de Krull de U(s\(\ell (3))\). (The Krull dimension of U(s\(\ell (3)))\)
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.
scientific article; zbMATH DE number 3957377
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