Zariski topologies on stratified spectra of quantum algebras

Abstract

A framework is developed to describe the Zariski topologies on the prime and primitive spectra of a quantum algebra $A$ in terms of the (known) topologies on strata of these spaces and maps between the collections of closed sets of different strata. A conjecture is formulated, under which the desired maps would arise from homomorphisms between certain central subalgebras of localized factor algebras of $A$. When the conjecture holds, spec $A$ and prim $A$ are then determined, as topological spaces, by a finite collection of (classical) affine algebraic varieties and morphisms between them. The conjecture is verified for ${\mathcal{O}}_q(G{L}_2(k))$, ${\mathcal{O}}_q(S{L}_3(k))$, and ${\mathcal{O}}_q(M_2(k))$ when $q$ is a non-root of unity and the base field $k$ is algebraically closed.

23 pages; 4 xypic diagrams

References in corpus (2)

• On the Spectra of Quantum Groups
• From quantum Ore extensions to quantum tori via noncommutative UFDs

Cited by in corpus (2)

• Prime factors of quantum Schubert cell algebras and clusters for quantum Richardson varieties
• The $q$-Division Ring, Quantum Matrices and Semi-classical Limits