On Ringel-Hall algebras

AbstractWe prove a comparison formula for the Donaldson–Thomas curve-counting invariants of two smooth and projective Calabi–Yau threefolds related by a flop. By results of Bridgeland any two such varieties are derived equivalent. Furthermore there exist pairs of categories of perverse coherent sheaves on both sides which are swapped by this equivalence. Using the theory developed by Joyce we construct the motivic Hall algebras of these categories. These algebras provide a bridge relating the invariants on both sides of the flop.

Download Full-text

Bridgeland's Hall algebras of tilted algebras and Lusztig's symmetries

Journal of Algebra ◽

10.1016/j.jalgebra.2016.07.021 ◽

2016 ◽

Vol 466 ◽

pp. 73-99 ◽

Cited By ~ 2

Author(s):

Haicheng Zhang

Keyword(s):

Hall Algebras ◽

Tilted Algebras

Download Full-text

Quantum Grothendieck rings and derived Hall algebras

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2013-0020 ◽

2013 ◽

Vol 0 (0) ◽

Cited By ~ 3

Author(s):

David Hernandez ◽

Bernard Leclerc

Keyword(s):

Hall Algebras ◽

Grothendieck Rings

Download Full-text

Spherical Hall Algebras of Weighted Projective Curves

International Mathematics Research Notices ◽

10.1093/imrn/rny158 ◽

2018 ◽

Vol 2020 (15) ◽

pp. 4721-4775

Author(s):

Jyun-Ao Lin

Keyword(s):

Vector Bundles ◽

Characteristic Functions ◽

Projective Curve ◽

Hall Algebra ◽

Smooth Projective Curve ◽

Coherent Sheaves ◽

Hall Algebras ◽

Arbitrary Genus ◽

Projective Curves ◽

Fixed Rank

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

Download Full-text