Why is the quantum Lorentz group not connected? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T01:29:45Z http://mathoverflow.net/feeds/question/86575 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/86575/why-is-the-quantum-lorentz-group-not-connected Why is the quantum Lorentz group not connected? Bob Yuncken 2012-01-24T21:35:30Z 2012-02-08T18:22:12Z <p>Podles and Woronowicz' construct the quantum Lorentz group, by which they mean $SL_q(2,\mathbb{C})$, as a quantum double of the compact quantum group $SU_q(2)$. More precisely, it is a bicrossed product of $K=SU_q(2)$ and its discrete quantum dual $\hat{K}$. So it seems that $SL_q(2,\mathbb{C})$ is not connected as a quantum topological space.</p> <p>Main Question: Why is the $q$-deformation of the connected group $SL(2,\mathbb{C})$ not connected?</p> <hr> <p>Here are some more precise questions:</p> <ol> <li><p>Given that $\hat{K}$ is supposed to be a $q$-analogue of the $AN$-component in the $KAN$-decomposition, is there any reasonable sense in which the discrete quantum dual of $SU_q(2)$ tends to the classical $AN$ group in the $q\to1$ limit?</p></li> <li><p>Are there alternative $q$-deformations of $SL(2,\mathbb{C})$ which remain connected?</p></li> <li><p>Is this strange non-connectedness related to the appearance of new "non-smooth" representations of $SL_q(2,\mathbb{C})$ in the work of Podles-Woronowicz, and Pusz?</p></li> </ol> http://mathoverflow.net/questions/86575/why-is-the-quantum-lorentz-group-not-connected/86619#86619 Answer by Nicola Ciccoli for Why is the quantum Lorentz group not connected? Nicola Ciccoli 2012-01-25T11:38:05Z 2012-01-25T17:42:21Z <p>I think that the assumption that the quantum Lorentz group is not connected is wrong, and this is due to some confusion about "duality", a word used, in this context, with different meanings. This is not special of $SL_2$.</p> <p>I partly share this confusion and I hope that my message will be corrected by someone with clearer ideas than I have.</p> <p>The compact group $K$ has a Pontryagin dual $\widehat{K}$. Having fixed on $K$ the standard Poisson-Lie structure, it also has a Poisson dual $K^*$ which is $AN$. the two things are patently not the same.</p> <p>Now the point is that on one hand the universal enveloping Lie algebra $u(\mathfrak{k})$ may be identified with a quantization of the group $AN$ (this is called quantum duality principle) and on the other hand is related (sorry, here I'm really "handwaving") to $\widehat{K}$ (guess to the convolution algebra). The keypoint is exactly this last relation, which is the right classical analogue of the quantum case.</p> <p>The relation between $AN$ and $\widehat{K}$ can be considered here a semiclassical effect.</p> <p>EDIT: Say $K$ is a compact Lie group and $R(K)$ its Lie algebra of representative functions. Then $R(K)$ is isomorphic to the group algebra of its Pontryagin dual $\mathbb C[\widehat{K}]$ (this can be seen as a version of Fourier transform).</p> <p>You may then say that the universal enveloping algebra $U(\mathfrak{k})$ is dual to $\mathbb C[\widehat{K}]$. </p> <p>On the other hand $U(\mathfrak k)$ is a quantization of the algebra of functions on the dual of the Lie algebra, $\mathfrak k^\ast$, with respect to the linear Lie-Poisson bracket.</p> <p>So the same object is related on one side to the discrete group $\widehat{K}$ and on the other side to the Poisson manifold $\mathfrak k^\ast$. Here there is no quantum group appearing, since we are assuming the trivial Poisson-Lie bracket on $K$.</p> <p>In the paper cited above we are assuming the standard Poisson-Lie structure on $K=SU(2)$. This implies that the Poisson manifold $\mathfrak{su}(2)^*$ has to be replaced by $AN$ with the dual Poisson-Lie group structure and that $U(\mathfrak k)$ gets replaced by $U_q(\mathfrak k)$. </p>