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Coproduct of modified Drinfeld-Cartan series for Yangians and quantum affine algebras in type A

Jérôme Milot

Abstract

We give explicit formulas for the coproducts of modified Drinfeld-Cartan generating series for the Yangian in type $A$ and for the quantum affine algebras in the particular type $A_2$. As an auxiliary result of the latter, we give an explicit presentation of positive prefundamental representations of the quantum affine algebra in the particular type $A_2$

Coproduct of modified Drinfeld-Cartan series for Yangians and quantum affine algebras in type A

Abstract

We give explicit formulas for the coproducts of modified Drinfeld-Cartan generating series for the Yangian in type and for the quantum affine algebras in the particular type . As an auxiliary result of the latter, we give an explicit presentation of positive prefundamental representations of the quantum affine algebra in the particular type
Paper Structure (12 sections, 11 theorems, 134 equations)

This paper contains 12 sections, 11 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Yangians, S-series, Theta series
  3. Yangians
  4. S-series and Theta series
  5. Formula for the coproduct of S-series in type A
  6. Quantum affine algebras, T-series and Theta series
  7. Quantum affine algebras
  8. T-series and Theta series
  9. Root vectors and universal R-matrix
  10. Positive prefundamental module
  11. Formula for the coproduct of T-series in type $A_2$
  12. Appendix : shifted Yangians and Theta series as associators

Key Result

Theorem 1.1

The Theta series for the quantum affine algebra $U_q(\hat{\mathop{\mathrm{\mathfrak{sl}}}\nolimits}_3)$ are given by the following two formulas where $\exp_q$ is a q-exponential and $\left[ x,y \right]_q = xy - qyx$ is a q-commutator.

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • remark 1
  • Definition 2.2
  • Theorem 2.3
  • remark 2
  • remark 3
  • Theorem 3.1
  • remark 4
  • ...and 21 more