Cohomological Hall algebras of 1-dimensional sheaves and Yangians
A seven-hour minicourse comprising seven one-hour lectures on cohomological Hall algebras of 1-dimensional sheaves and affine Yangians, with particular emphasis on their relationship through the derived McKay correspondence.
Instructor: Francesco Sala
Term: August 3–6, 2026
Location: MCM410, Morningside Center of Mathematics, Chinese Academy of Sciences
Time: August 3, 9:00–10:00 and 10:30–11:30; August 4, 9:00–10:00 and 10:10–11:10; August 6, 9:00–10:00, 10:30–11:30, and 14:30–15:30
Abstract
This series of seven one-hour lectures is based on joint papers with Diaconescu, Porta, Schiffmann, and Vasserot, arXiv:2502.19445 and arXiv:2603.03386. The main goal of the course is to explain the relationship between the cohomological Hall algebra (COHA) of 1-dimensional sheaves on the minimal resolution of a Kleinian singularity, the COHA of the preprojective algebra of the corresponding affine ADE quiver, and the corresponding affine Yangian via the derived McKay correspondence. This relationship is established using several key ingredients, including braid group actions on COHAs and Yangians, Bridgeland stability conditions, and the construction of “limiting” COHAs.
Programme
The minicourse is part of Mini courses on geometric representation theory and 3d mirror symmetry, held at the Morningside Center of Mathematics, Chinese Academy of Sciences.
Schedule
| Week | Date | Topic | Materials |
|---|---|---|---|
| Lecture 1 | August 3, 9:00–10:00 |
2-dimensional COHAs of quivers
Construction of 2-dimensional cohomological Hall algebras of quivers and their nilpotent variants in a broad setting. |
|
| Lecture 2 | August 3, 10:30–11:30 |
COHAs and multiparameter Yangians
The relationship between the 2-dimensional COHAs of quivers without edge loops and the corresponding multiparameter Yangians |
|
| Lecture 3 | August 4, 9:00–10:00 |
Braid group actions on COHAs and Yangians
Review of the Lie theory associated with quivers without edge loops; braid group actions on Yangians and their negative halves; and braid group actions on quiver COHAs via spherical twists. |
|
| Lecture 4 | August 4, 10:10–11:10 |
Affine Lie theory, affine Yangians, and COHAs
Lie theory of affine ADE quivers; finite, affine, and extended affine Weyl and braid groups; affine Yangians and their classical limits; and their relationship with quiver COHAs. |
|
| Lecture 5 | August 6, 9:00–10:00 |
Nilpotent COHAs of surfaces and derived McKay equivalence
Review of cohomological Hall algebras of smooth surfaces; introduction to nilpotent cohomological Hall algebras associated with pairs consisting of a smooth surface and a closed reduced subscheme; and introduction to the derived McKay correspondence from Van den Bergh’s perspective. |
|
| Lecture 6 | August 6, 10:30–11:30 |
Statement of the main result and limiting COHAs
Statement of the main result for the nilpotent COHA associated with the pair consisting of the minimal resolution of a Kleinian singularity and its exceptional locus; and introduction to the theory of limiting COHAs. |
|
| Lecture 7 | August 6, 14:30–15:30 |
Bridgeland stability conditions, limiting COHAs, and proof of the main result
Review of the theory of Bridgeland stability conditions; the relationship between limiting COHAs and Bridgeland stability conditions; and the use of limiting COHAs and extended affine braid group actions induced by derived equivalences to prove the main result. |