Spherical Hall algebras of a weighted projective curve

2014/10/03 by Jyun-Ao Lin, Lin, Jyun-Ao · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.1410.0896

openalex publication_date 2014/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we deal with the structure of the spherical Hall algebra of coherent sheaves with parabolic structures on a smooth projective curve of arbitrary genus. We provide a shuffle-like presentation of the vector bundle part and show the existence of the generic form. We also prove that the spherical Hall algebra contains the characteristic functions on all the Harder-Narasimhan strata. These results together imply Schiffmann's theorem on the existence of Kac polynomials for quasi-parabolic vector bundles of fixed rank and multi-degree over the curve. On the other hand, the shuffle structure we obtained is new and we make links to the representations of quantum affine algebras of type A.

Citations

• The Self-Duality Equations on a Riemann Surface

Cited by

• Higgs bundles and indecomposable parabolic bundles over the projective line

Related

• Spherical Hall algebras of curves and Harder-Narasimhan stratas
• The Hall algebra of the category of coherent sheaves on the projective line
• The Hall algebra of a curve
• Lectures on Hall algebras
• Noncommutative projective curves and quantum loop algebras