2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/71522Two 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.9 pagesRepresentation TheoryAlgebraic Geometry14L30; 17B37The bar automorphism in quantum groups and geometry of quiver representationstext