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The non-commutative n-torus appears in many applications of non-commutative geometry. To stay in the setting $n=2$: it is a C$^\ast$-algebra generated by unitaries $u$ and $v$, satisfying $u v = e^{i \theta} v u$. It is the deformation of the 2-torus, i.e. a group.

So my question is: besides viewing the nc torus as a 'non-commutative space', is it also a compact quantum group? That is, is there Hopf algebraic structure in it?

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3 Answers 3

The $C^*$-algebra versions are treated in this paper by Piotr Soltan:

http://arxiv.org/abs/0904.3019

The abstract reads: We prove that some well known compact quantum spaces like quantum tori and some quantum two-spheres do not admit a compact quantum group structure. This is achieved by considering existence of traces, characters and nuclearity of the corresponding $\mathrm{C}^*$-algebras.

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I don't know about the $C^*$-algebra version, but I can tell you about the algebraic version (the algebra generated by $u$ and $v$, invertible, such that $uv = qvu$).

It is not a Hopf algebra but a "braided group", that is, a Hopf algebra in some braided category (classical Hopf algebras being, in this parlance, "Hopf algebras in the category of vector spaces with the trivial twist"). Concretely, there is a map of algebras $A \to A \otimes A$ satisfying all the axioms you want, except that $A \otimes A$ is not made into an algebra in the way you think.

If I were allowed a bit of self-advertising, I'd recommend §4 of

http://arxiv.org/abs/0911.5287

Majid's book on quantum groups may have some formulae about the codiagonal in the quantum tori.

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Despite the negative result quoted by MTS, there have been some attempts to put a Hopf-like structure on the quantum torus.

One of these attemps, which seems orthogonal to the one mentioned by Pierre in his answer, is via Hopfish algebras. To be short, Hopfish algebras (after Tang-Weinstein-Zhu) are unital algebras equipped a coproduct, a counit and an antipode that are morphisms in the Morita category (they are bimodules,rather than actual algebra morphisms).

The Hopfish structure on the quantum torus has been studied in details in this paper.

To be complete, let me emphazise the following point (taken from the above paper):

It is important to note that, although the irrational rotation algebra may be viewed as a deformation of the algebra of functions on a 2-dimensional torus, our hopfish structure is not a deformation of the Hopf structure associated with the group structure on the torus. Rather, the classical limit of our hopfish structure is a second symplectic groupoid structure on $T^∗\mathbb{T}^2$ (...), whose quantization is the multiplication in the irrational rotation algebra. We thus seem to have a symplectic double groupoid which does not arise from a Poisson Lie group.

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