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$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs II: injectivity

Ming Lu, Shiquan Ruan

Abstract

We show that the morphism $Ω$ from the $\imath$quantum loop algebra $^{\texttt{Dr}}\widetilde{\mathbf{U}}(L\mathfrak{g})$ of split type to the $\imath$Hall algebra of the weighted projective line is injective if $\mathfrak{g}$ is of finite or affine type. As a byproduct, we use the whole $\imath$Hall algebra of the cyclic quiver $C_n$ to realise the $\imath$quantum loop algebra of affine $\mathfrak{gl}_n$.

$\imath$Hall algebras of weighted projective lines and quantum symmetric pairs II: injectivity

Abstract

We show that the morphism from the quantum loop algebra of split type to the Hall algebra of the weighted projective line is injective if is of finite or affine type. As a byproduct, we use the whole Hall algebra of the cyclic quiver to realise the quantum loop algebra of affine .
Paper Structure (26 sections, 30 theorems, 143 equations)

This paper contains 26 sections, 30 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Preliminaries
  4. $\imath$Quantum groups
  5. A Drinfeld type presentation
  6. Star-shaped graph
  7. Hall algebras
  8. Semi-derived Hall algebras and $\imath$Hall algebras
  9. Realization of $\imath$quantum groups via quivers
  10. Weighted projective lines
  11. $\imath$Hall algebras and $\imath$quantum loop algebras
  12. Root vectors
  13. The homomorphism $\Omega$
  14. $\imath$Hall algebras of cyclic quivers and $\widetilde{{\mathbf U}}^\imath_v(\widehat{\mathfrak{gl}}_n)$
  15. Ringel-Hall algebras and quantum groups
  16. ...and 11 more sections

Key Result

Lemma 2.1

For $i\in \mathbb{I}$, there exists an automorphism $\mathbf T_i$ of the $\mathbb Q(v)$-algebra $\widetilde{{\mathbf U}}^\imath$ such that $\mathbf T_i(\mathbb{K}_\mu) =\mathbb{K}_{s_i\mu}$ for $\mu\in \mathbb Z\mathbb{I}$, and for $j\in \mathbb{I}$. Moreover, $\mathbf T_i$$(i\in \mathbb{I})$ satisfy the braid group relations, i.e., $\mathbf T_i \mathbf T_j =\mathbf T_j \mathbf T_i$ if $c_{ij}=0$

Theorems & Definitions (54)

  • Lemma 2.1: LW21b; also cf. KP11BK20
  • Definition 2.2: $\imath$quantum loop algebras, LW20b
  • Theorem 2.3: LW20b
  • Lemma 2.4: LR21
  • Lemma 2.5: LW19a
  • Definition 2.6
  • Proposition 2.7: LRW20a
  • Proposition 2.8: LW20a
  • Theorem 2.9: LW20a
  • Proposition 2.10
  • ...and 44 more