Varieties for Modules of Quantum Elementary Abelian Groups

Abstract

We define a rank variety for a module of a noncocommutative Hopf algebra $A=\Lambda ⋊G$ 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.

30 pages, submitted