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Double-bosonization and Majid's conjecture (V): grafting of quantum groups

Hongmei Hu, Naihong Hu

TL;DR

The paper develops a multi-tensor extension of Majid's generalized double-bosonization to realize larger Drinfeld–Jimbo quantum groups by grafting two (or more) smaller quantum groups along a connecting node. It constructs a braided-categorical framework using weakly quasitriangular dual pairs and dually-paired braided groups, enabling explicit cross-relations and central extensions on a grafted tensor space. Concrete realizations are provided: a simply-laced Type A graft yielding $U_q(\mathfrak{sl}_{n+m})$ and a non-simply-laced Type F4 graft from $U_q(\mathfrak{sl}_3)\oplus U_q(\mathfrak{sl}_2)$, including $q$-Serre relations and basis analyses. The approach offers a uniform, algebraic method to construct new quantum groups from existing building blocks, with potential extension to affine and indefinite types, connecting to Majid's quantum-tree program.

Abstract

The goal of this paper is to propose and carry out a program which allows us to yield a larger target quantum group of Drinfeld-Jimbo type via grafting two given smaller ones. To this effect, we first set up some basics of a multi-tensor product theory of the generalized double-bosonization construction (see [HH2] for the latter). Such a grafting method, depending on the choice of grafting node in the Dynkin diagram, including rank-induction and type-crossing constructions as special cases, provides a one-stop resolving strategy of the generation question of Majid's quantum tree, which grows up from the root part $U_q(\mathfrak{sl}_2)$ and decorates with the quantum groups of various Dynkin diagrams (of finite types) as the branches.

Double-bosonization and Majid's conjecture (V): grafting of quantum groups

TL;DR

The paper develops a multi-tensor extension of Majid's generalized double-bosonization to realize larger Drinfeld–Jimbo quantum groups by grafting two (or more) smaller quantum groups along a connecting node. It constructs a braided-categorical framework using weakly quasitriangular dual pairs and dually-paired braided groups, enabling explicit cross-relations and central extensions on a grafted tensor space. Concrete realizations are provided: a simply-laced Type A graft yielding and a non-simply-laced Type F4 graft from , including -Serre relations and basis analyses. The approach offers a uniform, algebraic method to construct new quantum groups from existing building blocks, with potential extension to affine and indefinite types, connecting to Majid's quantum-tree program.

Abstract

The goal of this paper is to propose and carry out a program which allows us to yield a larger target quantum group of Drinfeld-Jimbo type via grafting two given smaller ones. To this effect, we first set up some basics of a multi-tensor product theory of the generalized double-bosonization construction (see [HH2] for the latter). Such a grafting method, depending on the choice of grafting node in the Dynkin diagram, including rank-induction and type-crossing constructions as special cases, provides a one-stop resolving strategy of the generation question of Majid's quantum tree, which grows up from the root part and decorates with the quantum groups of various Dynkin diagrams (of finite types) as the branches.
Paper Structure (13 sections, 13 theorems, 137 equations)

This paper contains 13 sections, 13 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Generalities on double-bosonization theory
  3. Some basic facts and notations
  4. Majid's weakly quasitriagular dual pair
  5. Dually-paired braided groups
  6. Generalized double-bosonization
  7. Multi-Tensor Product of Generalized Double-Bosonization
  8. Multi-tensor product of (co-)quasitriangular Hopf algebras
  9. Weakly quasitriangular dual pairs
  10. Multi-tensor product of generalized double-bosonization
  11. Applications: the grafting construction of two quantum groups
  12. Simply-laced case: Type $A$
  13. Non-simply-laced case: Type $F_4$

Key Result

Theorem 2.1

$($HH2$)$ Let $R_{VV}$ be the $R$-matrix corresponding to a minuscule irreducible representation $T_{V}$ of $U_{q}(\mathfrak{g})$. There exists a normalization constant $\lambda$ such that $\lambda R=R_{VV}$. Then the new quantum group $U=U(V^{\vee}(R^{\prime},R_{21}^{-1}),\widetilde{U_{q}^{ext}(\ma the co-unit is $\epsilon (e^{i})=\epsilon (f_{i})=0$, the co-product is Here, the factor $q_*-q_*^

Theorems & Definitions (19)

  • Remark 2.1
  • Theorem 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Lemma 3.3
  • Proposition 3.3
  • ...and 9 more