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Short star products for quantum symmetric pairs and applications

Stefan Kolb, Milen Yakimov

Abstract

We prove that the star product for quantum symmetric pair coideal subalgebras is short. We apply this result to obtain new conceptual proofs, from first principles, of several fundamental facts about quantum symmetric pairs. In particular, we establish the existence of the algebra anti-automorphism $σ_τ$ and of the bar involution, without making use of the quasi K-matrix. We give a new elementary proof of a conjecture by Balagović and Kolb, sometimes referred to as the fundamental lemma for quantum symmetric pairs. We obtain a conceptual formula expressing the tensor quasi K-matrix in terms of the much studied quasi R-matrix and the Letzter map. This also allows for a new independent proof of the intertwiner property of the quasi K-matrix.

Short star products for quantum symmetric pairs and applications

Abstract

We prove that the star product for quantum symmetric pair coideal subalgebras is short. We apply this result to obtain new conceptual proofs, from first principles, of several fundamental facts about quantum symmetric pairs. In particular, we establish the existence of the algebra anti-automorphism and of the bar involution, without making use of the quasi K-matrix. We give a new elementary proof of a conjecture by Balagović and Kolb, sometimes referred to as the fundamental lemma for quantum symmetric pairs. We obtain a conceptual formula expressing the tensor quasi K-matrix in terms of the much studied quasi R-matrix and the Letzter map. This also allows for a new independent proof of the intertwiner property of the quasi K-matrix.
Paper Structure (32 sections, 41 theorems, 213 equations, 1 figure)

This paper contains 32 sections, 41 theorems, 213 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Short star products
  3. Quantum symmetric pairs.
  4. Goal
  5. Results
  6. Application 1: The algebra automorphism $\sigma_\tau$
  7. Application 2: The bar involution for ${\mathcal{B}}_{\mathbf{c}}$
  8. Application 3: The fundamental lemma for quantum symmetric pairs
  9. Application 4: The quasi $K$-matrix for ${\mathcal{B}}_{\mathbf{c}}$
  10. Organization
  11. The Letzter map
  12. Background on quantized enveloping algebras
  13. Radford's biproduct for parabolic subalgebras
  14. Quantum symmetric pairs
  15. The quantum horospherical subalgebra
  16. ...and 17 more sections

Key Result

Proposition 2.1

The algebra ${\mathcal{R}}_X^{\pm, l}$ is a left $H$-module under the left adjoint action, and a left $H$-comodule under ${\varDelta}^{\pm,l}_H$. With these two structures, ${\mathcal{R}}_X^{\pm,l}$ is a Hopf algebra in the category of left Yetter-Drinfeld modules over $H$. We can form Radford's bip

Figures (1)

  • Figure 1: Implications between main results

Theorems & Definitions (87)

  • Proposition 2.1: a-Radford85
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Proposition 2.4
  • proof
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • ...and 77 more