## Morita equivalence of multidimensional noncommutative tori.(English)Zbl 0968.46060

An $$n$$-dimensional noncommutative torus is an associative algebra with involution having unitary generators $$U_1,\dots ,U_n$$ obeying the relations $$U_kU_j=e(\theta_{kj})U_jU_k,$$ where $$e(t)=e^{2\pi it}$$ and $$\theta$$ is antisymmetric matrix. The same name is used for $$C^*$$-algebra $$A_{\theta}$$ generated by $$U_1,\dots ,U_n.$$ Let $$O(n,n|{\mathbb R})$$ be a group of linear transformations of the space $${\mathbb R}^{2n}$$ preserving the quadratic form $$x_1x_{n+1}+\dots +x_nx_{2n}$$ and $$SO(n,n|{\mathbb R})$$ be a subgroup of $$O(n,n|{\mathbb R})$$ consisting of matrices with integer entries and determinant $$+1.$$ The action of $$O(n,n|{\mathbb R})$$ on the space $${\mathcal T}_n$$ of antisymmetric matrices is defined by the formula $$\theta '=(g_{11}\theta +g_{12})(g_{21}\theta +g_{22})^{-1},$$ where $$g=(g_{ij})_{ij=1,2}\in O(n,n|{\mathbb R})$$ and $$n\times n$$-matrices $$g_{ij}$$ satisfy $$g_{11}^tg_{21}+g_{21}^tg_{11}=g_{12}^tg_{22}+g_{22}^tg_{12}=0,$$ $$g_{11}^tg_{22}+g_{21}^tg_{12}=1.$$ (This action is defined only on the dense subset of $${\mathcal T}_n$$ where $$(g_{21}\theta +g_{22})$$ is invertible.) Let ${\mathcal T}_n^0=\{\theta \in {\mathcal T}_n: g\theta \text{ is defined for all }g\in SO(n,n|{\mathbb R})\}.$ $${\mathcal T}_n^0$$ is dense in $${\mathcal T}_n$$ and a simple calculation shows that $${\mathcal T}_n^0$$ is carried into itself by the action of $$SO(n,n|{\mathbb R}).$$ The main result of the paper is the following.
Theorem. For $$\theta \in {\mathcal T}_n^0$$ and $$g\in SO(n,n|{\mathbb R})$$ the noncommutative torus corresponding to $$g\theta$$ is Morita equivalent to the noncommutative torus corresponding to $$\theta.$$ This remains true for the smooth versions of the noncommutative tori.

### MSC:

 46L85 Noncommutative topology
Full Text:

### References:

 [1] Boca F. P., J. Reine Angew. Math. 492 pp 179– (1997) [2] DOI: 10.1090/conm/062/878397 · doi:10.1090/conm/062/878397 [3] DOI: 10.1007/BF02100097 · Zbl 0854.46057 · doi:10.1007/BF02100097 [4] Rieffel M. A., Pacific J. Math. 9 pp 3– (1981) [5] DOI: 10.4153/CJM-1988-012-9 · Zbl 0663.46073 · doi:10.4153/CJM-1988-012-9 [6] DOI: 10.1090/conm/105/1047281 · doi:10.1090/conm/105/1047281 [7] Schwarz A., Nuclear Phys. B 5 pp 720–
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.