Discrete Invariants of Koszul Artin-Schelter Regular Algebras of Dimension four

TL;DR

This paper computes and interprets geometric invariants of certain four-dimensional Koszul Artin-Schelter regular algebras, using computational algebra and representation theory, to distinguish these algebras through their discrete invariants.

Contribution

It introduces a novel approach combining computational algebra, representation theory, and algebraic geometry to compute and interpret invariants of these algebras.

Findings

Computed superpotentials for known algebras.

Identified geometric invariants as sections of line bundles.

Distinguished algebras using discrete invariants.

Abstract

We compute the superpotentials for known families of Koszul Artin-Schelter regular algebras of dimension four using Magma, and apply Schur-Weyl duality from representation theory to determine the relevant invariants. Through the Borel-Weil theorem, we interpret these invariants as sections of line bundles over partial flag varieties, resulting in geometric invariants that, in some cases, correspond to K3 surfaces. We compute discrete invariants of these geometric invariants and use them to distinguish algebras.