Cohomology of bimodules over enveloping algebras
type: Item
instance of: scholarly article
title: Cohomology of bimodules over enveloping algebras
zbMATH Open document ID: 0566.17005
DOI: 10.1007/BF01164161
author: Kenneth A. Brown
author: Thierry Levasseur
published in: Mathematische Zeitschrift
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/173589
review text: Let \({\mathfrak g}\) be a finite-dimensional Lie algebra over C with enveloping algebra U. Let M be a (U-U) bimodule, finitely generated on both sides. For \(i\geq 0\), \(Ext^ i_ U(M,U)\), evaluated for the left modules M and U, is a (U-U) bimodule via the right action of U on M and on U. The central result concerns the case when \(M=U/P\), where P is a prime ideal of height t of U and \({\mathfrak g}\) is solvable. Under these circumstances, \(Ext^ i_ U(U/P,U)=0\) for \(i<t\); and \(Ext^ t_ U(U/P,U)\) is isomorphic, as a left module, to a nonzero left ideal of U/P, and, as a right module, to a right ideal of \(U/\tau_{\lambda}(P)\), where \(\tau_{\lambda}\) is the winding automorphism of U obtained using a certain element \(\lambda\) of (\({\mathfrak g}/[{\mathfrak g},{\mathfrak g}])^*\) determined by P (for example if P is co-Artinian then \(\lambda =-trace ad {\mathfrak g}).\) To prove this result a partial quotient ring R is constructed with respect to a certain subset S(P) of \(U\setminus P\) [see the first author, Compos. Math. 53, 347-367 (1984; Zbl 0561.16003)]. The ring R has several properties which can be exploited, for example R/PR is the division ring of fractions of U/P, PR has a regular normalizing set of t generators, R has global dimension t and \(Ext^ i_ U(U/P,U)\otimes_ U R\cong Ext^ i_ R(R/PR,R)\) as bimodules. It then remains to prove that \(Ext^ i_ U(U/P,U)\) is left and right S(P)-torsion free and is finitely generated on both sides, the latter being a consequence of an unpublished result of I. N. Bernstein on Harish-Chandra modules M, namely that \(Ext^ i_ U(M,U)\) is a Harish-Chandra module. The authors give Bernstein's proof. Various applications and generalizations of the main result are given.
Mathematics Subject Classification ID: 17B35
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 17B55
Mathematics Subject Classification ID: 16Exx
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16P50
zbMATH DE Number: 3902868
zbMATH Keywords: solvable Lie algebra
zbMATH Keywords: vanishing of Ext
zbMATH Keywords: finite-dimensional Lie algebra
zbMATH Keywords: winding automorphism
zbMATH Keywords: bimodules
zbMATH Keywords: Harish-Chandra modules
reviewed by: Patrick F. Smith
MaRDI profile type: Publication
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OpenAlex ID: W2051261487
description: scientific article; zbMATH DE number 3902868