Varieties for Modules of Quantum Elementary Abelian Groups

We define a rank variety for a module of a noncocommutative Hopf algebra A = L rtimes GA = Lambda rtimes G where L = k[X₁, ..., X_m]/(X₁^l, ..., X_m^l), G = (mathbbZ/lmathbbZ)^mLambda = k[X_1, dots, X_m]/(X_1^{ell}, dots, X_m^{ell}), G = (mathbb{Z}/ellmathbb{Z})^m and char k does not divide ℓ, 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 Λ. When ℓ=2, rank varieties for Λ-modules were constructed by Erdmann and Holloway using the representation theory of the Clifford algebra. We show that the rank varieties we obtain for Λ-modules coincide with those of Erdmann and Holloway.