nLab noncommutative torus

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:

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

Related concepts

• fuzzy sphere

• fuzzy funnel

References

Original articles:

• Marc A. Rieffel, $C^\ast$-algebras associated with irrational rotations, Pacific Journal of Mathematics 93:2 (1981), 415–429 (pdf, doi:10.2140/pjm.1981.93.415, euclid:pjm/1102736269)

• Marc Rieffel, Albert Schwarz, Morita equivalence of multidimensional noncommutative tori, Int. J. Math. 10 (1999) 289-299 (arXiv:math/9803057)

• George A. Elliott and Hanfeng Li, Morita equivalence of smooth noncommutative tori, Acta Math. Volume 199, Number 1 (2007), 1-27 (euclid:acta/1485891908)

Review:

• Alain Connes, p. 55, p. 217, p. 356 of: Noncommutative Geometry, Academic Press, San Diego, CA, 1994 (ISBN:9780080571751, pdf)

• Wikipedia, Noncommutative torus

• John Baez, Noncommutative tori

Non-commutative tori understood as strict deformation quantizations:

• Marc A. Rieffel, Deformation quantization of Heisenberg manifolds, Communications in Mathematical Physics 122 (1989) 531–562 [doi:10.1007/BF01256492]