Double-bosonization and Majid's conjecture, (I): rank-induction of $ABCD$

TL;DR

This paper confirms Majid's conjecture that quantum groups of classical types can be inductively constructed from a basic node using double-bosonization, providing detailed constructions and examples for types A, B, C, D.

Contribution

The paper verifies Majid's rank-induction conjecture for classical types, detailing the inductive construction of quantum groups via double-bosonization.

Findings

Confirmed the inductive construction of $U_q(rak g)$ for $ABCD$ types.

Provided explicit examples illustrating the construction process.

Validated the initial node as $U_q(rak{sl}_2)$ for all classical types.

Abstract

Majid developed in \cite{majid3} his double-bosonization theory to construct $U_q(\mathfrak g)$ and expected to generate inductively not just a line but a tree of quantum groups starting from a node. In this paper, the authors confirm the Majid's first expectation (see p. 178 \cite{majid3}) through giving and verifying the full details of the inductive constructions of $U_q(\mathfrak g)$ for the classical types, i.e., the $ABCD$ series. Some examples in low ranks are given to elucidate that any quantum group of classical type can be constructed from the node corresponding to $U_{q}(\mathfrak{sl}_2)$.