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Varieties for Modules of Quantum Elementary Abelian Groups

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Abstract

We define a rank variety for a module of a noncocommutative Hopf algebra \(A = \Lambda \rtimes G\) where \(\Lambda = k[X_1, \dots, X_m]/(X_1^{\ell}, \dots, X_m^{\ell}), G = (\mathbb{Z}/\ell\mathbb{Z})^m\) 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.

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Correspondence to Julia Pevtsova.

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The first author was supported by NSF grant #DMS–0500946.

The second author was supported by the Alexander von Humboldt Foundation and by NSF grants #DMS–0422506 and #DMS-0443476.

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Pevtsova, J., Witherspoon, S. Varieties for Modules of Quantum Elementary Abelian Groups. Algebr Represent Theor 12, 567–595 (2009). https://doi.org/10.1007/s10468-008-9100-y

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