The half-twist for U_q(g) representations
  We introduce the notion of a half-ribbon Hopf algebra, which is a ribbon Hopf algebra along with a distinguished element $t$ corresponding to twisting a ribbon by 180 degrees (the half-twist). We show that U_q(g) is a (topological) half-ribbon Hopf algebra, but only if one uses a modified ribbon element. We then discuss some consequences of using this modified ribbon element. 
Noah Snyder and Peter Tingley