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Preprint Number 1278

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1278. Masanori Itai and Boris Zilber
A model theoretic Rieffel's theorem of quantum 2-torus

Submission date: 8 August 2017


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 R \setminus 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 = (a θ_1 + b)/(c θ_1 + d)
for some a, b, c d in 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 [IZ] is adequate to its original definition in terms of non-commutative geometry.

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Full text arXiv 1708.02615: pdf, ps.

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