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Universal $R$-matrix of double parameter quantum affine algebra $U_{q,Q}({\hat {sl_2}})$

Fengchang Li, Masatake Maruyama, Hiroyuki Yamane

Abstract

We give the explicit formula of the universal $R$-matrix of a double parameter (or two-parameter, or multi-parameter) quantum affine algebra of type ${\mathrm{A}}_1^{(1)}$. For $N$ with $q_{00}q_{01}$ being a primitive $N$-th root of unity, we introduce its $2N$-dimensional representation and explicitly calculate the $R$-matrix associated with it via the universal $R$-matrix.

Universal $R$-matrix of double parameter quantum affine algebra $U_{q,Q}({\hat {sl_2}})$

Abstract

We give the explicit formula of the universal -matrix of a double parameter (or two-parameter, or multi-parameter) quantum affine algebra of type . For with being a primitive -th root of unity, we introduce its -dimensional representation and explicitly calculate the -matrix associated with it via the universal -matrix.

Paper Structure

This paper contains 19 sections, 21 theorems, 165 equations.

Table of Contents

  1. Introduction
  2. Definition of $U_{q,Q}({\hat{sl_2}})$
  3. Positive part $U^+$ of $U_{q,Q}({\hat{sl_2}})$
  4. Root vectors $E_{k\delta+\alpha_0}$, $E_{k\delta+\alpha_1}$ and $E_{(k+1)\delta}$ with $k\geq 0$
  5. Calculation of $\Delta(E_{n\delta})$ and $\Delta(E_{n\delta+\alpha_i})$
  6. Modifications ${\tilde{E}}_{k\delta}$ of $E_{k\delta}$
  7. Calculation of $\Delta({\tilde{E}}_{n\delta})$
  8. Negative part $U^-$
  9. Root vectors $F_{k\delta+\alpha_0}$, $F_{k\delta+\alpha_1}$ and $F_{(k+1)\delta}$ with $k\geq 0$
  10. Modifications ${\tilde{F}}_{k\delta}$ of $F_{k\delta}$
  11. Coproduct for negative part
  12. Universal ${\mathrm{R}}$-matrix
  13. Drinfeld paring
  14. PBW-Theorem
  15. Main theorem
  16. ...and 4 more sections

Key Result

Lemma 3.1

There exists a unique ${\mathbb{C}}$-algebra anti-isomorphism $\psi:U^+\to U^+$ such that $\psi(E_0):=E_1$ and $\psi(E_1):=E_0$.

Theorems & Definitions (33)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • proof
  • Lemma 3.7
  • ...and 23 more