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Shifted affine iquantum groups of quasi-split ADE types

Kang Lu, Weiqiang Wang, Alex Weekes

Abstract

We formulate shifted affine iquantum groups of arbitrary quasi-split ADE types via Drinfeld presentations. We construct GKLO-type representations of shifted affine iquantum groups via algebras of difference operators, which allow us to construct truncated shifted affine iquantum groups. This provides a q-deformation of truncated shifted iYangians in our prior work arising as a quantization of affine Grassmannian islices.

Shifted affine iquantum groups of quasi-split ADE types

Abstract

We formulate shifted affine iquantum groups of arbitrary quasi-split ADE types via Drinfeld presentations. We construct GKLO-type representations of shifted affine iquantum groups via algebras of difference operators, which allow us to construct truncated shifted affine iquantum groups. This provides a q-deformation of truncated shifted iYangians in our prior work arising as a quantization of affine Grassmannian islices.

Paper Structure

This paper contains 22 sections, 5 theorems, 114 equations.

Table of Contents

  1. Introduction
  2. Shifted affine iquantum groups of quasi-split ADE type
  3. Affine iquantum groups
  4. A variant of Drinfeld presentation
  5. Shifted affine iquantum groups
  6. $\mathrm{i}$GKLO Representations of shifted affine $\mathrm{i}$quantum groups
  7. Ring of difference operators
  8. iGKLO representations
  9. Proof of Theorem \ref{['thmq']} for the split type
  10. Relations \ref{['rel:HH']} and \ref{['rel:BB4']}
  11. Relation \ref{['rel:HB']}
  12. The case $c_{ij}=0$
  13. The case $c_{ij}=2$
  14. The case $c_{ij}=-1$
  15. Relation \ref{['rel:BB5']}
  16. ...and 7 more sections

Key Result

Lemma 2.3

Assuming the validity of the relations HH--HB, the following equivalences hold.

Theorems & Definitions (13)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Theorem 3.1: iGKLO representations
  • Remark 3.2
  • Lemma 4.1
  • proof
  • ...and 3 more