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A quantum shuffle approach to admissible quantum affine (super-)algebra of types $A_1^{(1)}$ and $C(2)^{(2)}$ and their equitable presentations

Xin Zhong, Naihong Hu

TL;DR

The paper develops a unified, Catalan-word–driven framework for the positive parts of the admissible quantum affine algebra of type $A_1^{(1)}$ and the quantum affine superalgebra $C(2)^{(2)}$, embedding them into a $q$-shuffle (super)algebra to realize Damiani and Beck PBW bases in closed form. By introducing Catalan elements and a Rosso-type embedding, the authors translate root-vector constructions into explicit shuffle-algebra expressions, and establish an equitable presentation for the admissible affine algebra along with a minimal bosonization for the superalgebra. They also connect two prominent PBW bases via generating-function techniques and prove new Catalan-relations that govern the interaction of the Catalan elements under the $q$-shuffle product, exploiting a recursive Catalan definition tailored to the super case. The results yield a practical, computable bridge between PBW bases, shuffle algebras, and equitable/bosonized presentations, with potential extensions to $i$-quantum groups and Drinfeld realizations.

Abstract

In this study, we focus on the positive part $U_q^{+}$ of the admissible quantum affine algebra $\mathcal{U}_q(\widehat{\mathfrak{s l}_2})$, newly defined in \cite{HZ}, and the quantum affine superalgebra $U_q(C(2)^{(2)})$. Both of these algebras have presentations involving two generators, $e_α$ and $e_{δ-α}$, which satisfy the cubic $q$-Serre relations. According to the works of Hu-Zhuang and Khoroshkin-Lukierski-Tolstoy, there exist the Damiani and the Beck $PBW$ bases for these two (super)algebras. In this paper, we employ the $q$-shuffle (super)algebra and Catalan words to present these two bases in a closed-form expression. Ultimately, the equitable presentations of $\mathcal{U}_q(\widehat{\mathfrak{sl}_2})$ and the bosonization of $U_q(C(2)^{(2)})$ are presented.

A quantum shuffle approach to admissible quantum affine (super-)algebra of types $A_1^{(1)}$ and $C(2)^{(2)}$ and their equitable presentations

TL;DR

The paper develops a unified, Catalan-word–driven framework for the positive parts of the admissible quantum affine algebra of type and the quantum affine superalgebra , embedding them into a -shuffle (super)algebra to realize Damiani and Beck PBW bases in closed form. By introducing Catalan elements and a Rosso-type embedding, the authors translate root-vector constructions into explicit shuffle-algebra expressions, and establish an equitable presentation for the admissible affine algebra along with a minimal bosonization for the superalgebra. They also connect two prominent PBW bases via generating-function techniques and prove new Catalan-relations that govern the interaction of the Catalan elements under the -shuffle product, exploiting a recursive Catalan definition tailored to the super case. The results yield a practical, computable bridge between PBW bases, shuffle algebras, and equitable/bosonized presentations, with potential extensions to -quantum groups and Drinfeld realizations.

Abstract

In this study, we focus on the positive part of the admissible quantum affine algebra , newly defined in \cite{HZ}, and the quantum affine superalgebra . Both of these algebras have presentations involving two generators, and , which satisfy the cubic -Serre relations. According to the works of Hu-Zhuang and Khoroshkin-Lukierski-Tolstoy, there exist the Damiani and the Beck bases for these two (super)algebras. In this paper, we employ the -shuffle (super)algebra and Catalan words to present these two bases in a closed-form expression. Ultimately, the equitable presentations of and the bosonization of are presented.

Paper Structure

This paper contains 15 sections, 46 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Background and Motivation
  3. Goal
  4. Further Applications
  5. Advantage
  6. Definition and main conclusions for admissible quantum affine algebra $\mathcal{U}_q(\widehat{\mathfrak{s l}}_2)$
  7. quantum affine superalgebra $U_q(C(2)^{(2)})$ and $q$-shuffle superalgebra
  8. Quantum Affine Superalgebras $U_q(C(2)^{(2)})$
  9. $q$-shuffle superalgebra
  10. Connection bewteen $U_q^{+}(C(2)^{(2)})$ and $q$-shuffle superalgebra I
  11. Connection bewteen $U_q^{+}(C(2)^{(2)})$ and $q$-shuffle superalgebra II
  12. Two preliminary Lemmas
  13. The proof of Theorem \ref{['Main thm 2']}
  14. Some relations involving the Catalan elements coming from Theorems \ref{['Main thm 1']} & \ref{['Main thm 2']}
  15. Equitable presentation of a minimal bosonization of $U_q(C(2)^{(2)})$

Key Result

Theorem 2

HZ$\mathcal{U}_q(\widehat{\mathfrak{sl}_2})$ is neither isomorphic to the standard quantum affine algebra $U_q(\widehat{\mathfrak{s l}_2})$ of type $A_1^{(1)}$ nor to its co-opposite object $U_q^{c o p}(\widehat{\mathfrak{sl}_2})$ as pointed Hopf algebras (even as algebras).

Theorems & Definitions (85)

  • Definition 1
  • Theorem 2
  • Definition 3
  • Proposition 4
  • Proposition 5
  • proof
  • Definition 6
  • Theorem 7
  • Definition 8
  • Remark 9
  • ...and 75 more