Quantum groups and edge contractions

We study the behaviors of quantum groups under an edge contraction. We show that there exists an explicit embedding induced by an edge contraction operation. We further conjecture that this explicit embedding is a section of an explicit subquotient. This conjecture is proved when restricts to negative/positive half of a quantum group. The compatibility of the Hopf algebra structure of, and many other intrinsic structures associated with, a quantum group with the embedding and subquotient is studied along the way. The embedding phenomena are further observed in various representation theoretic objects such as Weyl groups and Chevalley groups.