On the generalized Clifford algebra of a monic polynomial

TL;DR

This paper investigates the structure of generalized Clifford algebras derived from specific monic polynomials, revealing conditions under which they form Azumaya algebras with centers related to elliptic curves and discussing their representations.

Contribution

It provides a complete structural analysis of generalized Clifford algebras for particular cases, including their Azumaya property and connection to elliptic curves, expanding understanding of their algebraic and geometric properties.

Findings

Algebras are Azumaya of rank nine in most cases

Centers are coordinate rings of affine elliptic curves

Representation theory is discussed over algebraically closed fields of characteristic zero

Abstract

In this paper we study the generalized Clifford algebra defined by Pappacena of a monic (with respect to the first variable) homogeneous polynomial $\Phi(Z,X_1,\dots,X_n)=Z^d-\sum_{k=1}^d f_k(X_1,\dots,X_n) Z^{d-k}$ of degree $d$ in $n+1$ variables over some field $F$. We completely determine its structure in the following cases: $n=2$ and $d=3$ and either $\operatorname{char}(F)=3$, $f_1=0$ and $f_2(X_1,X_2)=e X_1 X_2$ for some $e \in F$, or $\operatorname{char}(F) \neq 3$, $f_1(X_1,X_2)=r X_2$ and $f_2(X_1,X_2)=e X_1 X_2+t X_2^2$ for some $r,t,e \in F$. Except for a few exceptions, this algebra is an Azumaya algebra of rank nine whose center is the coordinate ring of an affine elliptic curve. We also discuss representations of arbitrary generalized Clifford algebras assuming the base field $F$ is algebraically closed of characteristic zero.