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

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

### Citations:

Zbl 0252.17004; Zbl 0478.17006