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Drinfeld Isomorphism for Novel Quantum Affine Algebra of Type $A_{1}^{(1)}$

Rushu Zhuang, Ge Feng, Naihong Hu

TL;DR

The work establishes an intrinsic Drinfeld realization, $U^{D}_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)$, for the novel two-parameter quantum affine algebra of type $A_{1}^{(1)}$, $U_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)$. It constructs Ω-invariant generating functions and root-vector data to define a current-type presentation with relations $(D1)$–$(D6)$, and proves a rigorous Drinfeld Isomorphism by producing mutually inverse homomorphisms $\Psi$ and $\Xi$ between the Drinfeld realization and the standard presentation. The results rely on braid-group automorphisms, an anti-automorphism, and the detailed interplay of real and imaginary root vectors to ensure that the two realizations are algebraically equivalent. This equivalence enables a unified framework for exploring representations and PBW-type bases in this two-parameter quantum affine setting, broadening the understanding of Drinfeld doubles and current realizations in nonstandard quantum groups.

Abstract

In this paper, we first review the definition of the novel quantum affine algebra \(U_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) of type \(A_{1}^{(1)}\) given in \cite{FHZ, HZhuang}. Furthermore, by introducing \(Ω\)-invariant generating functions, we construct the Drinfeld realization \(U^{D}_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) of this algebra, and prove that \(U_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) and \(U^{D}_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) are algebraically isomorphic, which is known as the Drinfeld Isomorphism.

Drinfeld Isomorphism for Novel Quantum Affine Algebra of Type $A_{1}^{(1)}$

TL;DR

The work establishes an intrinsic Drinfeld realization, , for the novel two-parameter quantum affine algebra of type , . It constructs Ω-invariant generating functions and root-vector data to define a current-type presentation with relations –, and proves a rigorous Drinfeld Isomorphism by producing mutually inverse homomorphisms and between the Drinfeld realization and the standard presentation. The results rely on braid-group automorphisms, an anti-automorphism, and the detailed interplay of real and imaginary root vectors to ensure that the two realizations are algebraically equivalent. This equivalence enables a unified framework for exploring representations and PBW-type bases in this two-parameter quantum affine setting, broadening the understanding of Drinfeld doubles and current realizations in nonstandard quantum groups.

Abstract

In this paper, we first review the definition of the novel quantum affine algebra \(U_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) of type \(A_{1}^{(1)}\) given in \cite{FHZ, HZhuang}. Furthermore, by introducing -invariant generating functions, we construct the Drinfeld realization \(U^{D}_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) of this algebra, and prove that \(U_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) and \(U^{D}_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)\) are algebraically isomorphic, which is known as the Drinfeld Isomorphism.
Paper Structure (9 sections, 23 theorems, 89 equations)

This paper contains 9 sections, 23 theorems, 89 equations.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Hopf algebra and Drinfeld double.
  4. Automorphisms of $U_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)$
  5. Definition of Root Vectors and Commutation Rules
  6. Drinfeld realization of $U_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)$
  7. Drinfeld Isomorphism
  8. Proof of Theorem 3.5
  9. Proof of Theorem 3.6

Key Result

Proposition 2.2

The algebra $U_{\textbf{q}}(\widehat{\mathfrak{sl}}_2)$ is a Hopf algebra with the comultiplication $\Delta$, the counit $\varepsilon$, the antipode $S$, satisfying the following relations:

Theorems & Definitions (38)

  • Definition 2.1: HZhuang, Def. 3.1
  • Proposition 2.2: HZhuang, Def. 3.1
  • Proposition 2.3: HZhuang, Prop. 2.3
  • Proposition 2.4: HZhuang, Prop. 2.4
  • Proposition 2.5: HZhuang, Def. 3.2
  • Proposition 2.6
  • Proposition 2.7
  • Theorem 2.8: HZhuang, Thm. 4.1
  • Corollary 2.9: HZhuang, Coro. 4.10
  • Proposition 2.10: HZhuang, Prop. 4.3
  • ...and 28 more