Alexandre Minets

Research

Papers/Preprints

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

Preprint. Last updated: October 2020.

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]

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]