Explicit Braid Group Reps from quantum SO(N) at roots of unity - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T07:01:37Z http://mathoverflow.net/feeds/question/27717 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/27717/explicit-braid-group-reps-from-quantum-son-at-roots-of-unity Explicit Braid Group Reps from quantum SO(N) at roots of unity Eric Rowell 2010-06-10T16:13:47Z 2010-06-10T16:13:47Z <p>This question is related to <a href="http://mathoverflow.net/questions/21161/is-the-pure-braid-group-on-three-strands-generated-as-a-normal-subgroup-of-the-br" rel="nofollow">this one</a> (and indeed the goals are similar).</p> <p>Let $N$ be odd and consider the braided fusion category $\mathcal{C}$ (actually modular) obtained from $U_q\mathfrak{so}_N$ at $q=e^{\pi i/(2N)}$ in the usual way (i.e. take the quotient of the category of tilting modules of Lusztig's integral form by the tensor ideal of negligible morphisms). This is sometimes denoted $SO(N)_2$ by physicists. Let $V$ denote the object in $\mathcal{C}$ analogous to the fundamental <em>spin</em> representation. Then the braid group $\mathcal{B}_n$ acts on the simple $End(V^{\otimes n})$-modules $Hom(W,V^{\otimes n})$ irreducibly, since in this case the image of $\mathcal{B}_n$ generates the centralizer algebras.</p> <p><strong>Question:</strong> how can one explicitly describe the braid group representations, say up to $n=5$? I.e. is there a uniform way of writing down the matrices of the generators of $\mathcal{B}_n$?</p> <p>What I know: </p> <ul> <li>For $q$ generic one can just use the $R$-matrix.</li> <li>For $N=3$ this is a (quotient of) the Temperley-Lieb algebras</li> <li>For $N=5$ this is a (quotient of) BMW-algebras (since $\mathfrak{so}_5$ is $\mathfrak{sp}_4$).</li> <li>For $N=7$ Westbury has a description MR2388243.</li> <li>For $N=k^2$ the category is integral and in fact group-theoretical</li> </ul> <p>So for $N\leq 7$ or a perfect square, one can get a useful description of the braid group action using the irreps of the quotient algebras or by the Drinfeld double construction. It is too much to ask for a description as a quotient of $\mathcal{B}_n$ for $q$ generic, but perhaps there is a way for this particular value of $q$.</p>