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

Ming Lu, Shiquan Ruan

Abstract

The $\imath$Hall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the $\imath$quantum loop algebra, which is a generalization of the $\imath$quantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The $\imath$Hall algebra of the $\imath$quiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $\imath$quantum group of split affine type A under the two presentations.

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

Abstract

The Hall algebra of a weighted projective line is defined to be the semi-derived Ringel-Hall algebra of the category of -periodic complexes of coherent sheaves on the weighted projective line over a finite field. We show that this Hall algebra provides a realization of the quantum loop algebra, which is a generalization of the quantum group arising from the quantum symmetric pair of split affine type ADE in its Drinfeld type presentation. The Hall algebra of the quiver algebra of split affine type A was known earlier to realize the same algebra in its Serre presentation. We then establish a derived equivalence which induces an isomorphism of these two Hall algebras, explaining the isomorphism of the quantum group of split affine type A under the two presentations.

Paper Structure

This paper contains 63 sections, 65 theorems, 351 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Quantum groups and Hall algebras
  4. $\imath$Quantum groups and $\imath$Hall algebras
  5. Goal
  6. Main results
  7. Comparison with previous works
  8. Further works
  9. Organization
  10. Acknowledgments
  11. A Drinfeld type presentation of $\imath$quantum groups
  12. The loop algebras and affine Lie algebras
  13. Affine $\imath$quantum groups of split type ADE
  14. A Drinfeld type presentation
  15. Star-shaped graph
  16. ...and 48 more sections

Key Result

Lemma 2.3

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 $a_{ij}=0$

Theorems & Definitions (124)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.3: LW21b; also cf. KP11BK20
  • Definition 2.4: $\imath$quantum loop algebras, LW20b
  • Lemma 2.5: LW20b
  • Theorem 2.6: LW20b
  • Lemma 2.7
  • proof
  • Lemma 3.1: LRW20a
  • Lemma 3.2: LRW20a
  • ...and 114 more