On the generalized Clifford algebra of a monic polynomial

In this paper we study the generalized Clifford algebra defined by Pappacena of a monic (with respect to the first variable) homogeneous polynomial Φ(Z,X1,…,Xn)=Zd−∑k=1dfk(X1,…,Xn)Zd−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 char(F)=3, f1=0 and f2(X1,X2)=eX1X2 for some e∈F, or char(F)≠3, f1(X1,X2)=rX2 and f2(X1,X2)=eX1X2+tX22 for some r,t,e∈F. Excluding 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.