  We define a rank variety for a module of a noncocommutative Hopf algebra $A =
\Lambda \rtimes G$ where $\Lambda = k[X_1, ..., X_m]/(X_1^{\ell}, ...,
X_m^{\ell})$, $G = ({\mathbb Z}/\ell{\mathbb Z})^m$, and $\text{char} k$ does
not divide $\ell$, in terms of certain subalgebras of $A$ playing the role of
"cyclic shifted subgroups". We show that the rank variety of a finitely
generated module $M$ is homeomorphic to the support variety of $M$ defined in
terms of the action of the cohomology algebra of $A$. As an application we
derive a theory of rank varieties for the algebra $\Lambda$. When $\ell=2$,
rank varieties for $\Lambda$-modules were constructed by Erdmann and Holloway
using the representation theory of the Clifford algebra. We show that the rank
varieties we obtain for $\Lambda$-modules coincide with those of Erdmann and
Holloway.

Quantum Algebra;Representation Theory