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Preprint Number 92

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92. Ivo Herzog, Sonia L'Innocente
The nonstandard quantum plane
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Submission date: 17 October 2007


Consider the quantum plane associated to a field k and defined as the k-algebra generated by the variables x and y modulo the relation yx =qxy, where the parameter q in k is not a root of unity. We can regard the quantum plane as a module over the quantized universal enveloping algebra U_q, related to the classical Lie algebra sl(2,k).
In the language of U_q-modules, we prove that the ring of definable scalars of the quantum plane is a von Neumann regular ring. We also investigate the closed set associated to the quantum plane in the Ziegler Spectrum of U_q.

Mathematics Subject Classification: 03C60

Keywords and phrases: Quantum plane, quantized universal enveloping algebra, definable scalars.

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