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On the center of the enveloping algebra of a classical simple Lie superalgebra. (English) Zbl 0885.17007

Let \(\mathfrak g\) be a complex semisimple Lie algebra with enveloping algebra \(U(\mathfrak g)\). Let \(Z(\mathfrak g)\) be the center of \(U(\mathfrak g)\). A classical result of Kostant states that there is an \(\text{ad }\mathfrak g\)-invariant subspace \(K\) of \(U(\mathfrak g)\), called the space of harmonics, such that the multiplication map \(K\otimes Z(\mathfrak g)\rightarrow U(\mathfrak g)\) is an isomorphism of \(\text{ad }\mathfrak g\)-modules. The proof depends heavily on the complete reducibility of any finite-dimensional \(\mathfrak g\)-module. In this paper the author proves the analogue of this result for the orthosymplectic Lie superalgebra \(\text{osp}(1,2r)\) (the only superalgebra, apart from Lie algebras, for which complete reducibility holds) and in general investigates the relationships between the centers of enveloping algebras of Lie superalgebras \(\mathfrak g_0+\mathfrak g_1\) and the corresponding centers for their even parts \(\mathfrak g_0\). He concentrates on the two cases of \(\text{osp}(1,2r)\) and \(\text{sl}(r,1)\) and generalizes earlier work of Pinczon on the case of \(\text{osp}(1,2)\). He raises such questions as whether the centralizer of \(\mathfrak g_0\) in \(U(\mathfrak g)\) is finitely generated, or commutative, or generated by \(Z(\mathfrak g)\) and \(Z(\mathfrak g_0)\), and whether minimal primitive ideals in \(U(\mathfrak g)\) are centrally generated, answering them in some cases.

MSC:

17B35 Universal enveloping (super)algebras
17A70 Superalgebras
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