A model theoretic Rieffel's theorem of quantum 2-torus
We defined a notion of quantum 2-torus $T_θ$ in "Masanori Itai and Boris Zilber, Notes on a model theory of quantum 2-torus $T_q^2$ for generic $q$, arXiv:1503.06045v1 [mathLO]" and studied its model theoretic property. In this note we associate quantum 2-tori $T_θ$ with the structure over ${\mathbb C}_θ= ({\mathbb C}, +, \cdot, y = x^θ),$ where $θ\in {\mathbb R} \setminus {\mathbb Q}$, and introduce the notion of geometric isomorphisms between such quantum 2-tori. We show that this notion is closely connected with the fundamental notion of Morita equivalence of non-commutative geometry. Namely, we prove that the quantum 2-tori $T_{θ_1}$ and $T_{θ_2}$ are Morita equivalent if and only if $θ_2 = {\displaystyle \frac{a θ_1 + b}{c θ_1 + d}}$ for some $ \left( \begin{array}{cc} a & b \\ c & d \end{array} \right) \in {\rm GL}_2({\mathbb Z})$ with $|ad - bc| = 1$. This is our version of Rieffel's Theorem in "M. A. Rieffel and A. Schwarz, Morita equivalence of multidimensional noncummutative tori, Internat. J. Math. 10, 2 (1999) 289-299" which characterises Morita equivalence of quantum tori in the same terms. The result in essence confirms that the representation $T_θ$ in terms of model-theoretic geometry \cite{IZ} is adequate to its original definition in terms of non-commutative geometry.