The Prime and the Primitive Spectra of a Quantum Bruhat Cell Translate

The prime spectra of two families of algebras, Sw, and w, w W, indexed by the Weyl group W of a semisimple finitely dimensional Lie algebra , are studied in the spirit of Joseph. The algebras Sw have been introduced by Joseph. They are q-analogues of the algebras of regular functions on w-translates of the open Bruhat cell of a semisimple Lie group G corresponding to the Lie algebra . We define a stratification of the spectra into components indexed by pairs (y1, y2) of elements of the Weyl group satisfying y1 ≤ w ≤ y2. Each component admits a unique minimal ideal which is explicitly described. We show the inclusion relation of closures to be that induced by Bruhat order.