# zbMATH — the first resource for mathematics

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 $${\mathfrak g}$$.

##### MSC:
 17B50 Modular Lie (super)algebras 17B35 Universal enveloping (super)algebras
Full Text: