# Alexandre Minets

## Research

### Papers/Preprints

KLR and Schur algebras for curves and semi-cuspidal representations (with Ruslan Maksimau)

International Mathematics Research Notices (2022)

Abstract: Given a smooth curve $$C$$, we define and study analogues of KLR algebras and quiver Schur algebras, where quiver representations are replaced by torsion sheaves on $$C$$. In particular, they provide a geometric realization for certain affinized symmetric algebras. When $$C = \mathbb P^1$$, a version of curve Schur algebra turns out to be Morita equivalent to the imaginary semi-cuspidal category of the Kronecker quiver in any characteristic. As a consequence, we argue that one should not expect to have a reasonable theory of parity sheaves for affine quivers.

[pdf]
[arXiv]
[journal]

Cohomological Hall algebras for Higgs torsion sheaves, moduli of triples and sheaves on surfaces

Selecta Mathematica (N.S.) Vol. 26 (2020), no. 2, paper No. 30 (67 pages)

Abstract: For any free oriented Borel–Moore homology theory $$A$$, we construct an associative product on the $$A$$-theory of the stack of Higgs torsion sheaves over a projective curve $$C$$. We show that the resulting algebra $$A\mathbf{Ha}_C^0$$ admits a natural shuffle presentation, and prove it is faithful when $$A$$ is replaced with usual Borel–Moore homology groups. We also introduce moduli spaces of stable triples, heavily inspired by Nakajima quiver varieties, whose $$A$$-theory admits an $$A\mathbf{Ha}_C^0$$-action. These triples can be interpreted as certain sheaves on $$\mathbb P(\omega_C\oplus \mathcal O_C)$$. In particular, we obtain an action of $$A\mathbf{Ha}_C^0$$ on the cohomology of Hilbert schemes of points on $$T^*C$$.

[pdf]
[arXiv]
[journal]

### Videos

Some recordings of me talking about math.

KLR and Schur algebras for curves and semi-cuspidal representations

a talk about arXiv:2010.01419 at the RepNet virtual seminar
[video]
[notes]

A primer on CoHas

a semi-expository talk at the ICMS Summer School
[video]
[notes]