Contents

# Contents

## Idea

A quotient of the torus, taken in the sense of noncommutative geometry.

## Definition

### As a Lie groupoid

The noncommutative torus with parameter $\theta \in \mathrm{U}(1)$ away from roots of unity is the groupoid quotient of the circle group $\mathrm{U}(1)$ by the action of the discrete group $\mathbf{Z}$ of integers, where the element $1 \in \mathbf{Z}$ acts by multiplication by $\theta$.

Here we can take $\theta = \exp(2\pi i \hbar)$, for $\hbar$ an irrational number (which, if we think of the non-commutative algebra here as arising from quantization, may be thought of as Planck's constant).

The following Lie groupoid is Morita equivalent to the above Lie groupoid:

take the groupoid quotient of $\mathrm{U}(1)^2$ (i.e., a torus) by the action of the Lie group $\mathbf{R}$ of real numbers such that

$a \cdot (u,v) \;=\; \big( u \exp(2\pi i a), v \exp(2\pi i a \hbar) \big)$

### As an algebra

Taking the groupoid algebra of the Lie groupoid constructed above, we obtain a noncommutative space in the sense of Connes noncommutative geometry.

For instance, in the realm of C*-algebras we get the C*-algebra generated by two unitary operators $U$ and $V$ subject to the relation

$U V=\theta V U.$

## Classification

The following classification is due to Rieffel 81.

If $\theta$ is a root of unity, then the resulting Lie groupoid is Morita equivalent to the smooth manifold $\mathrm{U}(1)$, i.e., the circle.

Otherwise, the noncommutative torus with parameter $\theta$ is not Morita equivalent to the circle (or any smooth manifold). Furthermore, two noncommutative tori with paremeters $\theta_1$ and $\theta_2$ are Morita equivalent if and only if $\theta_1$ and $\theta_2$ are in the same orbit with respect to the action of $\mathrm{GL}(2,\mathbf{Z})$ on $\mathrm{U}(1)$ defined as follows:

$\left[ \array{ a & b \\ c & d } \right] \cdot \exp(2\pi i \alpha) \;\coloneqq\; \exp \left( 2\pi i \frac {a\alpha + b} {c\alpha + d} \right) \,.$

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