Quantum groups and edge contractions

TL;DR

This paper investigates how quantum groups behave under edge contractions, establishing explicit embeddings, and explores their implications in representation theory, with proofs for certain cases and conjectures for the general scenario.

Contribution

It introduces explicit embeddings induced by edge contractions in quantum groups and proves the conjecture in specific cases, advancing understanding of their structural behavior.

Findings

Explicit embedding induced by edge contraction exists

Conjecture on the embedding being a section of a subquotient is proved in certain cases

Embedding phenomena observed in Weyl groups and Chevalley groups

Abstract

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.