noncommutative torus




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


As a Lie groupoid

The noncommutative torus with parameter θU(1)\theta \in \mathrm{U}(1) away from roots of unity is the groupoid quotient of the circle group U(1)\mathrm{U}(1) by the action of the discrete group Z\mathbf{Z} of integers, where the element 1Z1 \in \mathbf{Z} acts by multiplication by θ\theta.

Here we can take θ=exp(2πi)\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 U(1) 2\mathrm{U}(1)^2 (i.e., a torus) by the action of the Lie group R\mathbf{R} of real numbers such that

a(u,v)=(uexp(2πia),vexp(2πia)) 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 UU and VV subject to the relation

UV=θVU.U V=\theta V U.


The following classification is due to Rieffel 81.

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

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

[a b c d]exp(2πiα)exp(2πiaα+bcα+d). \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) \,.


Original articles:


Last revised on January 2, 2021 at 10:09:49. See the history of this page for a list of all contributions to it.