Virtual representation motives

Abstract

Principal $G{L}_n$-bundles (aka vector bundles) are locally trivial in the Zariski topology, whereas principal $PG{L}_n$-bundles (aka Azumaya algebras) are not, to the delight of every non-commutative algebraist. Still, this makes the calculation of motives of representation schemes of algebras next to impossible. In very special cases, Brauer-Severi schemes (and their motives) can be used to tackle this problem inductively. We illustrate this in the case of certain superpotential algebras.