# A note on generic Clifford algebras of binary cubic forms

## TL;DR

This paper investigates the algebraic structure of binary cubic generic Clifford algebras, revealing their properties as PI algebras, and characterizing when associated Clifford algebras are Azumaya, with explicit computations of key invariants.

## Contribution

It provides a detailed analysis of the representation theory, PI properties, and geometric invariants of binary cubic generic Clifford algebras, including criteria for Azumaya algebras.

## Key findings

- $ ext{Clifford algebra } \\mathcal C$ is an Artin-Schelter regular algebra of global dimension five.
- $ ext{Clifford algebra } \\mathcal C$ is a PI algebra of PI degree three.
- Explicit computation of point variety and discriminant ideals for $\\mathcal C$.

## Abstract

We study the representation theoretic results of the binary cubic generic Clifford algebra $\mathcal C$, which is an Artin-Schelter regular algebra of global dimension five. In particular, we show that $\mathcal C$ is a PI algebra of PI degree three and compute its point variety and discriminant ideals. As a consequence, we give a necessary and sufficient condition on a binary cubic form $f$ for the associated Clifford algebra $\mathcal C_f$ to be an Azumaya algebra.