A family of new Borel subalgebras of quantum groups

TL;DR

This paper constructs and classifies a new family of Borel subalgebras in quantum groups, revealing their structure and representation theory, including explicit examples for Uq(sl4).

Contribution

It introduces a novel family of Borel subalgebras with one-dimensional irreducible representations and proves their maximality and classification within quantum groups.

Findings

Constructed a new family of Borel subalgebras with triangular decomposition.

Proved these subalgebras have all irreducible representations one-dimensional.

Classified all triangular Borel subalgebras for Uq(sl4).

Abstract

We construct a family of right coideal subalgebras of quantum groups, which have the property that all irreducible representations are one-dimensional, and which are maximal with this property. The obvious examples for this are the standard Borel subalgebras expected from Lie theory, but in a quantum group there are many more. Constructing and classifying them is interesting for structural reasons, and because they lead to unfamiliar induced (Verma-)modules for the quantum group. The explicit family we construct in this article consists of quantum Weyl algebras combined with parts of a standard Borel subalgebra, and they have a triangular decomposition. Our main result is proving their Borel subalgebra property. Conversely we prove under some restrictions a classification result, which characterizes our family. Moreover we list for Uq(sl4) all possible triangular Borel subalgebras, using our underlying results and additional by-hand arguments. This gives a good working example and puts our results into context.