Explicit Descriptions of \$q-SO(2)\$, and \$q-Sp(2)\$? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T13:03:37Z http://mathoverflow.net/feeds/question/110757 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/110757/explicit-descriptions-of-q-so2-and-q-sp2 Explicit Descriptions of \$q-SO(2)\$, and \$q-Sp(2)\$? Ago Szekeres 2012-10-26T14:40:34Z 2012-10-28T21:03:55Z <p>When ever I hear noncommutative geometers talking about quantum groups, it is usually \$q-SU(2)\$ that they are discussing. As a result there are many good and explicit generator and relation presentations of this Hopf algebra. For an easy example take this other M.O. <a href="http://mathoverflow.net/questions/10581/kontsevich-and-geometric-quantization-and-the-podles-sphere" rel="nofollow">question</a>. I am curious to see what the simple examples of the other quantum groups series are. More specifically, could anyone give me a generator and relation description of the Hopf algebras \$\$ q-SO(2), ~~~~~~~~~~~~~~~~~~~ q-Sp(2)? \$\$ I know that somehow these are derivable from the some quantized enveloping algebra dual quantum groups, but that's a little too difficult for a ''classical'' geometer like me! </p> http://mathoverflow.net/questions/110757/explicit-descriptions-of-q-so2-and-q-sp2/110937#110937 Answer by Carlo Beenakker for Explicit Descriptions of \$q-SO(2)\$, and \$q-Sp(2)\$? Carlo Beenakker 2012-10-28T20:20:46Z 2012-10-28T21:03:55Z <p>If I am not mistaken, \$SO(2)\approx U(1)\$ has no nontrivial quantum deformation, but \$SO(3)\$ does; this is explicitly constructed in:</p> <p><A HREF="http://xxx.lanl.gov/abs/hep-th/9402069" rel="nofollow">Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups</A>, P. Podles (1994)</p> <p><A HREF="http://arxiv.org/abs/0810.0398" rel="nofollow">Quantum SO(3) groups</A>, P.M. Soltan (2008)</p> <p>For <em>q-Sp(2)</em>, see Section 3.2 of</p> <p><A HREF="http://www.math.ru.nl/~waltervs/laparesuIMRN.pdf" rel="nofollow">Noncommutative families of instantons</A>, G. Landi et al. (2008).</p> <p>More general references on quantum classical groups:</p> <p><A HREF="http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&amp;vol=28&amp;iss=1&amp;rank=5" rel="nofollow">Quantum deformation of classical groups</A>, T. Hayashi (1992)</p> <p><A HREF="http://arxiv.org/abs/math/9503225" rel="nofollow">Quantum symmetric spaces and related q-orthogonal polynomials</A>, M. Noumi and T. Sugitani (1995)</p> <p><A HREF="http://arxiv.org/abs/math/0511618" rel="nofollow">Orthogonal and symplectic quantum matrix algebras and Cayley-Hamilton theorem for them</A>, O. Ogievetsky and P. Pyatov (2005)</p>