Cohomology of bimodules over enveloping algebras
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is left and right S(P)-torsion free and is finitely generated on both sides, the latter being a consequence of an unpublished result of I. N. Bernstein on Harish-Chandra modules M, namely that \(Ext^ i_ U(M,U)\) is a Harish-Chandra module.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.
scientific article; zbMATH DE number 3902868
scientific article