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Invariant theory of $\imath$quantum groups of type AIII

Li Luo, Zheming Xu

Abstract

We develop an invariant theory of quasi-split $\imath$quantum groups $\mathbf{U}_n^\imath$ of type AIII on a tensor space associated to $\imath$Howe dualities. The first and second fundamental theorems for $\mathbf{U}_n^\imath$-invariants are derived.

Invariant theory of $\imath$quantum groups of type AIII

Abstract

We develop an invariant theory of quasi-split quantum groups of type AIII on a tensor space associated to Howe dualities. The first and second fundamental theorems for -invariants are derived.
Paper Structure (15 sections, 13 theorems, 67 equations)

This paper contains 15 sections, 13 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Quantum groups and $\imath$quantum groups
  3. Preliminary on Hopf algebras
  4. Quantum general linear group
  5. Quantum coordinate algebra $\mathcal{V}_{N,M}$
  6. Review of the FFT and the SFT for $\mathbf{U}_N$
  7. $\imath$quantum group $\mathbf{U}_n^\imath$
  8. $\imath$Schur duality
  9. $\imath$Howe duality
  10. The dual space $\mathcal{V}_{n,m,d}^{\imath *}$
  11. Invariant theory for $\mathbf{U}_n^\imath$
  12. $\mathbf{U}_n^\imath$-invariants
  13. The composition map $\Phi_{p,r,d}$
  14. The FFT for $\mathbf{U}_n^\imath$
  15. The SFT for $\mathbf{U}_n^\imath$

Key Result

Lemma 2.2

Let $H$ be a Hopf algebra, and let $A$ (resp. $B$) be a left (resp. right) $H$-module algebra.

Theorems & Definitions (28)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Theorem 2.4: LZZ11Z20
  • Remark 2.5
  • Lemma 2.6
  • Lemma 2.7
  • Theorem 2.8
  • ...and 18 more