Vertex embeddings of quantum groups via quivers - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T02:20:33Z http://mathoverflow.net/feeds/question/100747 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100747/vertex-embeddings-of-quantum-groups-via-quivers Vertex embeddings of quantum groups via quivers Peter McNamara 2012-06-27T07:15:05Z 2012-06-27T07:15:05Z <p>Let U<sub>q</sub> be a quantised enveloping algebra of type affine ADE (untwisted). By the loop presentation of U<sub>q</sub>, we see that for each vertex of the finite Dynkin diagram, there is an inclusion U<sub>q</sub>($\hat{sl_2}$)&rarr;U<sub>q</sub>.</p> <p>Now let us restrict to the positive part U<sup>+</sup> of U<sub>q</sub>. It is well known how to construct U<sup>+</sup> from the representations of the affine Dynkin diagram (given some orientation so that it becomes a quiver). This proceeds either by the Hall algebra approach or Lusztig's geometric approach with perverse sheaves.</p> <p>My question is: Can we see the appearance of the vertex embeddings discussed in the first paragraph via the quiver perspective?</p> <p>The obvious approach of trying to choose an orientation of our affine quiver so that it has a full subcategory (with objects of the correct dimensions) equivalent to the category of representations of the Kronecker quiver doesn't seem to work (eg look at E7 and the vertex of valence 1 closest to the central node).</p>