Bimodules sur une algèbre de Lie résoluble. (Bimodules over a solvable Lie algebra)
Let k be an algebraically closed field of characteristic 0, \({\mathfrak g}\) be a solvable Lie algebra of finite dimension over k and U be its enveloping algebra. If M is a left U-module, denote by d(M) the Gelfand- Kirillov dimension of M. For a U-U-bimodule M, set K(M) for the Krull dimension of M as a left U-module and K'(M) for the same on the right. If A is a k-algebra, let \(s(A)=\inf \{d(M)\); M an A-module \(\neq 0\}\). For \(f\in {\mathfrak g}*\), define: \({\mathfrak g}^ f=\{x\in {\mathfrak g}\); f([x,\({\mathfrak g}])=0\}\). Finally, let E be the set of non-zero semi- invariants of U/P (P being a prime ideal of U), it satisfies the necessary (Ore) conditions for localization: let \((U/P)_ E\) be the localized. Three main theorems are proved in this paper: Theorem 1: If P is a prime ideal of U then \(d(U/P)=K(U/P)+s(U/P).\) Theorem 2: If M is a U-U-bimodule of finite type on each side then \(K(M)=K'(M).\) (These 2 theorems had been obtained previously by \textit{K. A. Brown} and \textit{P. F. Smith} [Q. J. Math., Oxf. II. Ser. 36, 129-139 (1985; Zbl 0573.17010)] under the extra hypothesis that \({\mathfrak g}\) is ad-algebraic. To carry over their proof to the general case, two key ingredients are used: results of J. McConnell and P. Tauvel about the structure of \((U/P)_ E\) and the introduction of the notion of an ''excellent'' primitive ideal). Theorem 3: One has \(2s((U/I(f))_ E)=\dim {\mathfrak g}-\dim {\mathfrak g}^ f\) where I(f) is the primitive ideal associated to f by the Dixmier map. (This answers a question of P. Tauvel. One should point out that by this theorem, the integer dim \({\mathfrak g}^ f\) is for the first time connected with the enveloping algebra of \({\mathfrak g}.)\) Several lemmas and corollaries with their own interest are also proved. Many counterexamples show that some results cannot be improved.
scientific article; zbMATH DE number 3998051
scientific article