| Summary: | <p>In this thesis, we consider <em>q</em>-deformations of multiplicative Hypertoric varieties, where <em>q</em>∈&Kopf;<sup>x</sup> for &Kopf; an algebraically closed field of characteristic 0. We construct an algebra 𝒟<sub>q</sub> of <em>q</em>-difference operators as a Heisenberg double in a braided monoidal category. We then focus on the case where <em>q</em> is specialized to a root of unity. In this setting, we use 𝒟<sub>q</sub> to construct an Azumaya algebra on an <em>l</em>-twist of the multiplicative Hypertoric variety, before showing that this algebra splits over the fibers of both the moment and resolution maps. Finally, we sketch a derived localization theorem for these Azumaya algebras. </p>
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