The Fine Moduli Space of Representations of Clifford Algebras

TL;DR

This paper constructs fine moduli spaces for irreducible representations of Clifford algebras associated with binary forms, linking algebraic representations to geometric moduli spaces of stable vector bundles on a related curve.

Contribution

It introduces a method to parametrize irreducible representations of Clifford algebras via moduli spaces of stable vector bundles on a specific algebraic curve.

Findings

Constructed fine moduli spaces for irreducible representations of Clifford algebras.

Connected algebraic representations to geometric moduli spaces of vector bundles.

Provided a geometric framework for understanding representations of Clifford algebras.

Abstract

Given a fixed binary form $f(u,v)$ of degree $d$ over a field $k$, the associated \emph{Clifford algebra} is the $k$-algebra $C_f=k\{u,v\}/I$, where $I$ is the two-sided ideal generated by elements of the form $(\alpha u+\beta v)^{d}-f(\alpha,\beta)$ with $\alpha$ and $\beta$ arbitrary elements in $k$. All representations of $C_f$ have dimensions that are multiples of $d$, and occur in families. In this article we construct fine moduli spaces $U=U_{f,r}$ for the irreducible $rd$-dimensional representations of $C_f$ for each $r \geq 2$. Our construction starts with the projective curve $C \subset \mathbb{P}^{2}_{k}$ defined by the equation $w^d=f(u,v)$, and produces $U_{f,r}$ as a quasiprojective variety in the moduli space $\mathcal{M}(r,d_r)$ of stable vector bundles over $C$ with rank $r$ and degree $d_r=r(d+g-1)$, where $g$ denotes the genus of $C$.