TITLE: Is the nc torus a quantum group?
QUESTION [9 upvotes]: 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?

REPLY [4 votes]: 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.