Moduli of linear representations, symmetric products and the non commutative Hilbert scheme

TL;DR

This survey explores the relationships between moduli spaces of algebra representations, non-commutative Hilbert schemes, and symmetric products, extending classical results to positive characteristic and non-commutative contexts.

Contribution

It introduces a map connecting these moduli spaces, generalizes the Hilbert-Chow morphism to non-commutative settings, and discusses extensions of non-commutative desingularization and D-brane theories.

Findings

Established a map generalizing the Hilbert-Chow morphism

Extended non-commutative desingularization results to positive characteristic

Suggested applications to non-commutative D-brane models

Abstract

In this survey paper we study the relationships between the coarse moduli space which parameterizes the finite dimensional linear representations of an associative alegebra, the non commutative hilbert scheme and the affine scheme which is the spectrum of the abelianization of algebra of the divided powers. In particular we will show a map which specialize to the Hlibert - Chow morphism when the associative algebra is commutative. The extension to the positive characteristic case of some of the results due to L. Le Bruyn on noncommutative desingularization is outlined. The possibility to use our construction to extend the work done by C.H.Liu and S.T.Yau on D-Branes to their more recent work on non commutative case is underlined.