# On the generalized Clifford algebra of a monic polynomial

Research paper by **Adam Chapman, Jung-Miao Kuo**

Indexed on: **08 Jun '14**Published on: **08 Jun '14**Published in: **Mathematics - Rings and Algebras**

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#### 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.