Abstract

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The topological structure of the primitive spectrum of a generic quantized coordinate 

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ring A is known "piecewise". Assuming the base field is algebraically closed, there is 

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an algebraic torus H acting on A in such a way that there are only finitely many H-orbits 
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in prim A, each of which is homeomorphic to an affine variety (actually, another torus). 

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The Zariski topology on prim A is determined by the topologies on the H-orbits together 

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with data relating the closed sets in different H-orbits. There is a parallel picture 

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determining the Zariski topology on spec A. We will discuss how to frame the required 

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data in terms of maps from the H-orbits to auxiliary spaces. Analysis of examples such 

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as quantum GL_2 leads to the conjecture that these auxiliary data can be expressed as 

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morphisms between affine varieties.