Varieties for Modules of Quantum Elementary Abelian Groups

2006/03/16 by Julia Pevtsova, Pevtsova, Julia, Sarah Witherspoon +1 · 1 citation
Mathematics · #16E40 #16W30 #FOS: Mathematics #Quantum Algebra (math.QA) #Representation Theory (math.RT) #math.QA #math.RT #msc:16E40 #msc:16W30

paper · pdf · doi:10.48550/arxiv.math/0603409

30 pages, submitted

arxiv created 2006/03/16 · arxiv updated 2009/12/01

Abstract

We define a rank variety for a module of a noncocommutative Hopf algebra A = Λ\rtimes G where Λ= k[X₁, ..., X_m]/(X₁^ℓ, ..., X_m^ℓ), G = (\mathbb Z/ℓ\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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