A model theoretic Rieffel's theorem of quantum 2-torus

Abstract

We defined a notion of quantum 2-torus $T_{\theta }$ 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 $T_{\theta }$ with the structure over ${\mathbb{C}}_{\theta }=(\mathbb{C},+,\cdot ,y=x^{\theta }),$ where $\theta \in \mathbb{R}\setminus \mathbb{Q}$, 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 $T_{{\theta }_1}$ and $T_{{\theta }_2}$ are Morita equivalent if and only if ${\theta }_2=\frac{a{\theta }_1+b}{c{\theta }_1+d}$ for some $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in {\mathrm{GL}}_2(\mathbb{Z})$ 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 $T_{\theta }$ in terms of model-theoretic geometry \cite{IZ} is adequate to its original definition in terms of non-commutative geometry.

10 pages; Mathematical Logic and foundations

References in corpus (2)

• The theory of exponential sums
• Notes on a model theory of quantum 2-torus for generic q