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The Prime Spectrum and Representation Theory of Generalized Weyl Algebras, with Applications to Quantized Algebras

Abstract

This work develops the theory of generalized Weyl algebras (GWAs) in order to study generic quantized algebras. The ideas behind the classification of simple modules over GWAs are used to describe the noncommutative prime spectrum for certain GWAs. The primary example studied is the 2x2 reflection equation algebra A in the case that q is not a root of unity. Simple finite dimensional A-modules are classified, finite dimensional weight modules are shown to be semisimple, Aut(A) is computed, and the prime spectrum of A is computed along with its Zariski topology. It is shown that A satisfies the Dixmier-Moeglin equivalence and that it satisfies a Duflo-type theorem to some extent. The notion of a Poisson GWA is developed and used to explore the semiclassical limit of A. For some other quantized algebras and their classical counterparts, the GWA theory is demonstrated as a means to study the prime spectrum.

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