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$\imath$Hall algebra of the projective line and $q$-Onsager algebra

Ming Lu, Shiquan Ruan, Weiqiang Wang

Abstract

The $\imath$Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of coherent sheaves on the projective line. This $\imath$Hall algebra is shown to realize the universal $q$-Onsager algebra (i.e., $\imath$quantum group of split affine $A_1$ type) in its Drinfeld type presentation. The $\imath$Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism of the $q$-Onsager algebra under the two presentations.

$\imath$Hall algebra of the projective line and $q$-Onsager algebra

Abstract

The Hall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of -periodic complexes of coherent sheaves on the projective line. This Hall algebra is shown to realize the universal -Onsager algebra (i.e., quantum group of split affine type) in its Drinfeld type presentation. The Hall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these two Hall algebras, explaining the isomorphism of the -Onsager algebra under the two presentations.

Paper Structure

This paper contains 22 sections, 37 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. $\imath$Hall algebra of the projective line
  3. Coherent sheaves on ${\mathbb P}^1_{\Bbbk}$
  4. Hall algebras
  5. Category of $1$-periodic complexes
  6. $\imath$Hall algebras
  7. The $q$-Onsager algebra and its current presentation
  8. The $q$-Onsager algebra
  9. A Drinfeld type presentation of $\widetilde{{\mathbf U}}^\imath$
  10. $\imath$Hall algebra and $q$-Onsager algebra
  11. The homomorphism $\Omega$
  12. Relation \ref{['iDR3']}
  13. Relation \ref{['eq:hh1']} and Jordan $\imath$quiver
  14. Relation \ref{['eq:hB1']}
  15. The product $\widehat{\Theta}_{m}*[{\mathcal{O}}(r)]$
  16. ...and 7 more sections

Key Result

Lemma \oldthetheorem

For any acyclic complex $K^\bullet$ and $p \ge 2$, we have

Theorems & Definitions (68)

  • Remark \oldthetheorem
  • Lemma \oldthetheorem: also cf. LinP
  • proof
  • Lemma \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Lemma \oldthetheorem: LW22
  • Lemma \oldthetheorem
  • ...and 58 more