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Centers of reduced enveloping algebras. (English) Zbl 0932.17020
The main result of this paper is a computation of the inverse image of a functional in the Zassenhaus variety.
Let \({\mathfrak g}\) be a classical semisimple Lie algebra over an algebraically closed field of positive characteristic. The spectrum of \(Z({\mathfrak g})\), the center of the universal enveloping algebra \(U= U({\mathfrak g})\), is called the Zassenhaus variety of \({\mathfrak g}\) and denoted \({\mathcal Z}\). It is naturally mapped onto \({\mathfrak g}^{*(1)}\), the Frobenius twist of the dual space of \({\mathfrak g}\). The inverse image scheme of any \(\chi\in{\mathfrak g}^{*(1)}\) is explicitly described (Theorem 10) under a mild restriction on the characteristic. The algebra of functions \(Z_\chi\) on this inverse image is isomorphic to a direct sum of certain partial coinvariant algebras.
This computation is used to describe the category of representations of a regular functional. The reduced enveloping algebra at a regular functional is the matrix algebra \(M_q(Z_\chi)\) where \(q= p^m\) and \(m\) is the number of positive roots of \({\mathfrak g}\).

17B50 Modular Lie (super)algebras
17B35 Universal enveloping (super)algebras
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