Varieties for Modules of Quantum Elementary Abelian Groups

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We define a rank variety for a module of a noncocommutative Hopf algebra A = Λ G where Λ= k[X₁,..., Xₘ]/(X₁ℓ,..., Xₘℓ), G = (Z/ℓZ)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.