Around 1990, Ringel famously introduced an algebraic structure on the space of functions over the moduli space of objects in finitary categories during his study of quantum groups. This algebra is now called the Ringel-Hall algebra.

Recently, an analogous construction of the algebraic structure on the homology of the moduli space of objects in an abelian category has been intensively studied. Such an algebra is called the cohomological Hall algebra (CoHA). For an abelian category with homological dimension less than or equal to two, Kapranov and Vasserot constructed the CoHA using derived algebraic geometry. For an abelian category with homological dimension three and a Calabi-Yau structure, Kontsevich and Soibelman conjectured that one can construct the CoHA using the critical cohomology of the moduli space, and they verified this for the category of representations over the Jacobi algebra associated with a quiver with potential.

In this talk, I will explain a general construction of the cohomological Hall algebra associated with 3-Calabi-Yau categories and how it recovers all known constructions of the CoHA. If time permits, I will explain how this construction can be used to define a bialgebra structure on the homology of the moduli space of objects in 2-Calabi-Yau categories. This talk is based on joint work (https://arxiv.org/abs/2406.12838) with Hyeonjun Park and Pavel Safronov, as well as on another collaboration in progress with Ben Davison.

Place: Room 206, RIMS