BIMSA >
Disquisitions on Monoidal Categories and Operads
Geometric construction of quiver tensor products
Geometric construction of quiver tensor products
Organizers
Speaker
Daigo Ito
Time
Monday, March 2, 2026 4:00 PM - 5:00 PM
Venue
A3-1-301
Online
Zoom 559 700 6085
(BIMSA)
Abstract
By a classical theorem of Beilinson, the perfect derived category $\operatorname{Perf}(\mathbb{P}^n)$ of projective space is equivalent to the category of derived representations of a certain quiver with relations. The vertex-wise tensor product of quiver representations corresponds to a symmetric monoidal structure $\otimes_{\mathrm{quiv}}$ on $\operatorname{Perf}(\mathbb{P}^n)$. We demonstrate that this symmetric monoidal structure can be geometrically described as an “extended convolution product.” Specifically, the Fourier--Mukai kernel is given by the closure of the torus multiplication map in $(\mathbb{P}^n)^3$. We further extend the result to any smooth proper toric variety of Bondal--Ruan type. Under toric mirror symmetry, this extended convolution product corresponds to the tensor product of constructible sheaves on a real torus.