Given an affine algebraic variety V and a quantization Oq(V ) of its coordinate ring, it is conjectured that the primitive ideal space of Oq(V ) can be expressed as a topological quotient of V . Evidence in favor of this conjecture is discussed, and positive solutions for several types of varieties (obtained in joint work with E. S. Letzter) are described. In particular, explicit topological qu...

We describe the prime and primitive spectra for quantized enveloping algebras at roots of 1 in characteristic zero in terms of the prime spectrum of the underlying enveloping algebra. Our methods come from the theory of Hopf algebra crossed products. For primitive ideals we obtain an analogue of Duflo’s Theorem, which says that every primitive ideal is the annihilator of a simple highest weight...

We extend two well-known results on primitive ideals in enveloping algebras of semisimple Lie algebras, the Irreducibility theorem and Duflo theorem, to much wider classes of algebras. Our general version of Irreducibility theorem says that if A is a positively filtered associative algebra such that grA is a commutative Poisson algebra with finitely many symplectic leaves, then the associated v...

For any countable directed graph E we describe the primitive ideal space of the corresponding generalized Cuntz-Krieger algebra C∗(E).

Let G = GLn(q) be the general linear group of degree n ≥ 2 defined over a finite field Fq of characteristic p. We fix a prime l 6= p and let stand R for a local principal ideal domain having characteristic 0, maximal ideal lR, and containing a primitive p-th root of unity. Then the residue field K = R/lR has characteristic l and a primitive p-th root of unity. By a Steinberg lattice of G over R...

The separating space of a derivation onA is a separating ideal [2, Chapter 5]; it also satisfies the same property for the left products. The following assertions are of the most famous conjectures about derivations on Banach algebras: (C1) every derivation on a Banach algebra has a nilpotent separating ideal; (C2) every derivation on a semiprime Banach algebra is continuous; (C3) every derivat...