The bar automorphism in quantum groups and geometry of quiver   representations
  Two geometric interpretations of the bar automorphism in the positive part of a quantized enveloping algebra are given. The first is in terms of numbers of rational points over finite fields of quiver analogues of orbital varieties; the second is in terms of a duality of constructible functions provided by preprojective varieties of quivers. 
Philippe Caldero, Markus Reineke