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Zbl 0917.17006
Polo, Patrick
Dynkin diagrams and enveloping algebras of semisimple Lie algebras. (Diagrammes de Dynkin et algèbres enveloppantes d'algèbres de Lie semi-simples.) (French)
[J] Ann. Sci. Éc. Norm. Supér. (4) 31, No. 5, 631-657 (1998). ISSN 0012-9593

Let $\frak g$ be a complex semisimple Lie algebra with enveloping algebra $U$. Let $A$ be a minimal primitive quotient of $U$ with a finite-dimensional quotient. The purpose of this paper is to refine and extend to $A$ certain rigidity results about $U$ that were proved in a previous paper of {\it J. Alev} and the author [Adv. Math. 111, 208-226 (1995; Zbl 0823.17011)]. \par More precisely, the author shows that the Dynkin diagram of $\frak g$ is a Morita invariant not only of $A$ but also of its rings of invariants by finite groups of automorphisms, and the flag variety of $\frak g$ is determined within all generalized flag varieties by its ring of differential operators. He derives two consequences: if $U$ is isomorphic to the enveloping algebra $U'$ of another reductive Lie algebra $\frak g'$, then $\frak g\cong\frak g'$; and any automorphism of $U$ acts on its center (thus also on a dense open subset of its primitive spectrum) as a diagram automorphism. He conjectures that any automorphism of $U$ in fact acts on its entire primitive spectrum as a diagram automorphism. He also gives $K$-theoretic versions of both this rigidity result and its conjectural extension. Finally, he shows under the weaker hypothesis that the central character of $A$ is regular that at least the Weyl group of $\frak g$ is a Morita invariant of $A$. \par The methods of this paper are largely those of the earlier one cited above, but a recent result of Soergel is also invoked.
[W.M.McGovern (Seattle)]

MSC 2000:: *17B35 Universal enveloping algebras (Lie algebras)
16S30 Universal enveloping algebras of Lie algebras (associative)

Keywords: Dynkin diagram; enveloping algebra; Morita invariant; semisimple Lie algebra

Citations: Zbl 0823.17011

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