Morita equivalence of primitive factors of \(U(\mathbf s\mathbf l (2))\). (English) Zbl 0814.17008

Hodges, Timothy J.

Deodhar, Vinay (ed.), Kazhdan-Lusztig theory and related topics. Proceedings of an AMS special session, held May 19-20, 1989 at the University of Chicago, Lake Shore Campus, Chicago, IL, USA. Providence, RI: American Mathematical Society. Contemp. Math. 139, 175-179 (1992).

Let \(U=U({\mathfrak {sl}} (2,\mathbb{C}))\) be the enveloping algebra of the Lie algebra \({\mathfrak {sl}} (2,\mathbb{C})\). The minimal primitive ideals of \(U\) are of the form \((\Omega- \alpha)\) where \(\alpha\in \mathbb{C}\) and \(\Omega\) is the Casimir element. In [J. Algebra 24, 551-564 (1973; Zbl 0252.17004)] J. Dixmier proved that the algebras \(B_ \alpha= U/ (\Omega- \alpha)\) are non-isomorphic as \(\mathbb{C}\)-algebras. On the other hand, J. T. Stafford [Math. Proc. Camb. Philos. Soc. 91, 29-37 (1982; Zbl 0478.17006)] showed that many of the \(B_ \alpha\) are Morita equivalent via “translation functors” and asked whether it was true that all the simple \(B_ \alpha\) are equivalent. In this note we answer this question by showing that two such algebras are equivalent only if there is a translation functor defining equivalence. In passing we give a new and very short proof of Dixmier’s result. (From the paper).
For the entire collection see [Zbl 0784.00017].

Reviewer: P.M.Cohn (London)

MSC:

17B35

Universal enveloping (super)algebras

Keywords:

primitive factors; universal enveloping algebra; Morita equivalence; Hattori-Stallings map; Casimir element; translation functor

Citations:

Zbl 0252.17004; Zbl 0478.17006