Is the nc torus a quantum group? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-22T00:49:50Z http://mathoverflow.net/feeds/question/76913 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/76913/is-the-nc-torus-a-quantum-group Is the nc torus a quantum group? Anonymous 2011-10-01T08:00:50Z 2011-12-13T15:13:06Z <p>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.</p> <p>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?</p> http://mathoverflow.net/questions/76913/is-the-nc-torus-a-quantum-group/76918#76918 Answer by Pierre for Is the nc torus a quantum group? Pierre 2011-10-01T12:20:27Z 2011-10-05T07:15:03Z <p>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 $uv = qvu$). </p> <p>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.</p> <p>If I were allowed a bit of self-advertising, I'd recommend §4 of</p> <p><a href="http://arxiv.org/abs/0911.5287" rel="nofollow">http://arxiv.org/abs/0911.5287</a></p> <p>Majid's book on quantum groups may have some formulae about the codiagonal in the quantum tori.</p> http://mathoverflow.net/questions/76913/is-the-nc-torus-a-quantum-group/76936#76936 Answer by MTS for Is the nc torus a quantum group? MTS 2011-10-01T17:40:34Z 2011-10-01T17:40:34Z <p>The $C^*$-algebra versions are treated in this paper by Piotr Soltan:</p> <p><a href="http://arxiv.org/abs/0904.3019" rel="nofollow">http://arxiv.org/abs/0904.3019</a></p> <p>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. </p> http://mathoverflow.net/questions/76913/is-the-nc-torus-a-quantum-group/83343#83343 Answer by DamienC for Is the nc torus a quantum group? DamienC 2011-12-13T15:13:06Z 2011-12-13T15:13:06Z <p>Despite the negative result quoted by MTS, there have been some attempts to put a Hopf-like structure on the quantum torus. </p> <p>One of these attemps, which seems orthogonal to the one mentioned by Pierre in his answer, is <em>via</em> <a href="http://arxiv.org/abs/math/0510421" rel="nofollow">Hopfish algebras</a>. 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). </p> <p>The Hopfish structure on the quantum torus has been studied in details in <a href="http://arxiv.org/abs/math/0604405" rel="nofollow">this paper</a>. </p> <p>To be complete, let me emphazise the following point (taken from the above paper): </p> <blockquote> <p>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.</p> </blockquote>